A new lower bound for the chromatic number of general Kneser hypergraphs

TL;DR

This paper introduces a new combinatorial parameter called equitable colorability defect that improves lower bounds for the chromatic number of general Kneser hypergraphs, surpassing previous bounds and ensuring the existence of colorful subhypergraphs.

Contribution

It defines the equitable colorability defect and establishes a new, stronger lower bound for the chromatic number of general Kneser hypergraphs, improving upon prior bounds by Ziegler and Dol'nikov-Kríz.

Findings

The new lower bound always matches or exceeds Ziegler's bound.

Several hypergraph families show the new bound can be arbitrarily larger than previous bounds.

The existence of colorful subhypergraphs in any proper coloring is guaranteed.

Abstract

A general Kneser hypergraph ${\rm KG}^r(\mathcal{H})$ is an $r$-uniform hypergraph that somehow encodes the edge intersections of a ground hypergraph $\mathcal{H}$. The colorability defect of $\mathcal{H}$ is a combinatorial parameter providing a lower bound for the chromatic number of ${\rm KG}^r(\mathcal{H})$ which is addressed in a series of works by Dol'nikov [Sibirskii Matematicheskii Zhurnal, 1988}], K\v{r}\'{\i}\v{z} [Transaction of the American Mathematical Society, 1992], and Ziegler~[Inventiones Mathematicae, 2002]. In this paper, we define a new combinatorial parameter, the equitable colorability defect of hypergraphs, which provides some common improvements of these works. Roughly speaking, we propose a new lower bound for the chromatic number of general Kneser hypergraphs which substantially improves Ziegler's lower bound. It is always as good as Ziegler's lower bound and we provide several families of hypergraphs for which the difference between these two lower bounds is arbitrary large. This specializes to a substantial improvement of the Dol'nikov-K\v{r}\'{\i}\v{z} lower bound for the chromatic number of general Kneser hypergraphs as well. Furthermore, we prove a result ensuring the existence of a colorful subhypergraph in any proper coloring of general Kneser hypergraphs which strengthens Meunier's result [The Electronic Journal of Combinatorics, 2014].

Equations

$$E(\operatorname{KG}^{r}(n,k))=\left\{\{e_{1},\ldots,e_{r}\}\colon|e_{i}|=k,e_{i}\subseteq[n],\mbox{ and }e_{i}\cap e_{j}=\varnothing\mbox{ for each $i\neq j\in[r]$}\right\}.$$

$$\chi ({\mathrm{KG}}^r(n,k))=\lceil \frac{n-r(k-1)}{r-1}\rceil .$$

$$E({\mathrm{KG}}^r(\mathcal{H}))=\left\{\{e_1,...,e_r\}:e_i\in E(\mathcal{H})\text{ and }{e}_i\cap e_j=\varnothing \text{ for each }i\ne j\in [r]\right\}.$$

$$\chi \left({\mathrm{KG}}^r(\mathcal{H})\right)\ge \lceil \frac{{\mathrm{cd}}^r(\mathcal{H})}{r-1}\rceil .$$

$$\chi ({\mathrm{KG}}^r(\mathcal{H}))\ge \lceil \frac{{\mathrm{ecd}}^r(\mathcal{H})}{r-1}\rceil .$$

$$E(\mathcal{H})=\left\{e\subseteq [n]:|e|=k\text{mbox}ande⊈\{n-a+1,\dots ,n\}\right\}.$$

$$E(\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H}))=\Big{\{}\{e_{1},\ldots,e_{r}\}\colon e_{1},\ldots,e_{r}\in E(\mathcal{H})\text{ and }\{e_{1},\ldots,e_{r}\}\text{ is $\boldsymbol{s}$-disjoint }\Big{\}}.$$

$$\operatorname{cd}_{\boldsymbol{s}}^{r}(\mathcal{H})=\bar{n}-\max\left\{\displaystyle\sum_{j=1}^{r}|N_{j}|\colon N_{j}\subseteq[n],\;\{N_{1},\ldots,N_{r}\}\mbox{ is $\boldsymbol{s}$-disjoint and $E(\mathcal{H}[N_{j}])=\varnothing$ for each }j\in[r]\right\},$$

$$\chi ({\mathrm{KG}}_{\mathbf{s}}^r(\mathcal{H}))\ge \lceil \frac{{\mathrm{cd}}_{\mathbf{s}}^r(\mathcal{H})}{r-1}\rceil ,$$

$$\displaystyle\operatorname{ecd}^{r}_{\boldsymbol{s}}(\mathcal{H})=\bar{n}-\max\left\{\displaystyle\sum_{j=1}^{r}|N_{j}|\colon N_{j}\subseteq[n],\{N_{1},\ldots,N_{r}\}\mbox{ is equitable $\boldsymbol{s}$-disjoint and $E(\mathcal{H}[N_{j}])=\varnothing$ for each }j\in[r]\right\}.$$

$$\chi ({\mathrm{KG}}_{\mathbf{s}}^r(\mathcal{H}))\ge \lceil \frac{{\mathrm{ecd}}_{\mathbf{s}}^r(\mathcal{H})}{r-1}\rceil .$$

$$\Big{\{}F\subseteq V(C)\uplus V(K)\colon F\neq\varnothing,F\cap V(C)\in C\cup\{\varnothing\}\text{ and }F\cap V(K)\in K\cup\{\varnothing\}\Big{\}}.$$

$$\left|\left\{\epsilon \in Z_p:(\epsilon ,i)\in A\right\}\right|\le s_i.$$

$$l(\tau )=p\cdot h(\tau )+|\{\epsilon \in Z_p:|{\tau }^{\epsilon }|>h(\tau )\}|.$$

$${\tau }_{\epsilon }=\left\{j\in [n]:(\epsilon ,j)\in \tau \right\}.$$

$$l(\mathcal{H})=\max \left\{l(\tau ):\tau \in K\text{ such that }E(\mathcal{H}[{\tau }_{\epsilon }])=\varnothing \text{mbox}foreach\epsilon \in Z_p\right\}.$$

$$m=l(\mathcal{H})+\max\big{\{}l(\tau_{c})\colon\tau\in K\mbox{ and }l(\tau)>l(\mathcal{H})\big{\}},$$

$$\tau_{c}=\big{\{}(\varepsilon,c(e))\in Z_{p}\times[t]\colon e\in E(\mathcal{H})\mbox{ and }e\subseteq\tau_{\varepsilon}\big{\}}.$$

$$\bar{\tau }=\{\epsilon \in Z_p:{\tau }^{\epsilon }=\varnothing \}\in {\sigma }_{p-2}^{p-1}.$$

$$\bar{\tau }=\bigcup _{\{\epsilon \in Z_p:|{\tau }^{\epsilon }|=h(\tau )\}}{\tau }^{\epsilon }\in W.$$

$${\bar{\tau }}_c=\{\epsilon \in Z_p:{{\tau }_c}^{\epsilon }=\varnothing \}\in {\sigma }_{p-2}^{p-1}.$$

$${\bar{\tau }}_c=\bigcup _{\{\epsilon \in Z_p:|{{\tau }_c}^{\epsilon }|=h({\tau }_c)\}}{{\tau }_c}^{\epsilon }\in U.$$

$${\epsilon }_1=s_0(\{\epsilon \in Z_p:{\tau }^{\epsilon }=\varnothing \})\ne s_0(\{\epsilon \in Z_p:{{\tau }^{\prime }}^{\epsilon }=\varnothing \})={\epsilon }_2,$$

$$l({\tau }^{\prime })=p-|\{\epsilon \in Z_p:{{\tau }^{\prime }}^{\epsilon }=\varnothing \}|>p-|\{\epsilon \in Z_p:{\tau }^{\epsilon }=\varnothing \}|=l(\tau ),$$

$$l(\tau )=p\cdot h(\tau )+|\{\epsilon \in Z_p:|{\tau }^{\epsilon }|>h(\tau )\}|\text{mbox}andl({\tau }^{\prime })=p\cdot h({\tau }^{\prime })+|\{\epsilon \in Z_p:|{{\tau }^{\prime }}^{\epsilon }|>h({\tau }^{\prime })\}|.$$

$$\max \left\{|\{\epsilon \in Z_p:|{\tau }^{\epsilon }|>h(\tau )\}|,|\{\epsilon \in Z_p:|{{\tau }^{\prime }}^{\epsilon }|>h({\tau }^{\prime })\}|\right\}\le p-1$$

$$|\{\epsilon \in Z_p:|{\tau }^{\epsilon }|>h\}|=|\{\epsilon \in Z_p:|{{\tau }^{\prime }}^{\epsilon }|>h\}|.$$

$${\epsilon }_1=s_1(\bigcup _{\{\epsilon \in Z_p:|{\tau }^{\epsilon }|=h\}}{\tau }^{\epsilon })\ne s_1(\bigcup _{\{\epsilon \in Z_p:|{{\tau }^{\prime }}^{\epsilon }|=h\}}{{\tau }^{\prime }}^{\epsilon })={\epsilon }_2,$$

$$\bigcup _{\{\epsilon \in Z_p:|{\tau }^{\epsilon }|=h\}}{\tau }^{\epsilon }\ne \bigcup _{\{\epsilon \in Z_p:|{{\tau }^{\prime }}^{\epsilon }|=h\}}{{\tau }^{\prime }}^{\epsilon }.$$

$$\left\{\epsilon \in Z_p:|{{\tau }^{\prime }}^{\epsilon }|=h\right\}⊊\left\{\epsilon \in Z_p:|{\tau }^{\epsilon }|=h\right\},$$

Full text

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1 Introduction and Main Results

For two positive integers $n$ and $k$, the two symbols $[n]$ and ${[n]\choose k}$ respectively stand for the set $\{1,\ldots,n\}$ and the family of all $k$-subsets of $[n]$. A hypergraph $\mathcal{H}$ is a pair $(V(\mathcal{H}),E(\mathcal{H}))$ where $V(\mathcal{H})$ - the vertex set - is a finite set and $E(\mathcal{H})$ - the edge set - is a family of nonempty subsets of $V(\mathcal{H})$. For an integer $r\geq 2$, the hypergraph $\mathcal{H}$ is called $r$-uniform if each of its edges has the cardinality $r$. For a set $N\subseteq V(\mathcal{H})$, the induced subhypergraph $\mathcal{H}[N]$ is the hypergraph whose vertex set is $N$ and whose edge set is $\{e\in E(\mathcal{H})\colon e\subseteq N\}$.

Let $n,k,$ and $r$ be positive integers, where $r\geq 2$ and $n\geq rk$. The Kneser hypergraph $\operatorname{KG}^{r}(n,k)$ is an $r$-uniform hypergraph with vertex set ${[n]\choose k}$ and edge set consisting of all $r$-tuples of pairwise disjoint members of ${[n]\choose k}$, i.e.,

[TABLE]

In a break-through, Lovász [15] determined the chromatic number of Kneser graphs $\operatorname{KG}^{2}(n,k)$ and solved a long-standing conjecture posed by Kneser [12]. His proof gave birth to an area of combinatorics which nowadays is known as topological combinatorics. Mainly, this area of combinatorics is focused on the study of coloring properties of graphs and hypergraphs by using algebraic topological tools. Alon, Frankl, and Lovász [4] extended Lovász’s result to Kneser hypergraphs $\operatorname{KG}^{r}(n,k)$ by proving

[TABLE]

This result gave an affirmative answer to a conjecture posed by Erdős [10] as well. Above-mentioned results were generalized in many ways. One of the most promising generalizations is the one found by Dol’nikov [9], extended by Kříž [13, 14], and generalized by Ziegler [23, 24]. To state these results, we first need to introduce some preliminary notation and definitions.

Let $\mathcal{H}$ be a hypergraph and $r$ an integer where $r\geq 2$. The general Kneser hypergraph $\operatorname{KG}^{r}(\mathcal{H})$ is a hypergraph with vertex set $E(\mathcal{H})$ and edge set

[TABLE]

For $r=2$, we would use $\operatorname{KG}(\mathcal{H})$ rather than $\operatorname{KG}^{2}(\mathcal{H})$. Note that the Kneser hypergraph $\operatorname{KG}^{r}(n,k)$ can be obtained in this way by setting $\mathcal{H}=\left([n],{[n]\choose k}\right)$. The $r$-colorability defect of $\mathcal{H}$, denoted by $\operatorname{cd}^{r}(\mathcal{H})$, is the minimum number of vertices that must be removed from $\mathcal{H}$ so that the induced subhypergraph on the remaining vertices is $r$-colorable. Dol’nikov [9] (for $r=2$) and Kříž [13, 14] proved that

[TABLE]

If $\mathcal{H}=\left([n],{[n]\choose k}\right)$, then this result implies $\operatorname{KG}^{r}(n,k)\geq\left\lceil{n-r(k-1)\over r-1}\right\rceil$ concluding the Alon-Frankl-Lovász result [4] (one can color $\operatorname{KG}^{r}(n,k)$ by $\left\lceil{n-r(k-1)\over r-1}\right\rceil$ colors).

The equitable $r$-colorability defect of $\mathcal{H}$, denoted by $\operatorname{ecd}^{r}(\mathcal{H})$, is the minimum number of vertices that must be removed from $\mathcal{H}$ so that the induced subhypergraph on the remaining vertices has an equitable $r$-coloring. Recall that a hypergraph has an equitable $r$-coloring if it admits a proper $r$-coloring such that the sizes of its color classes differ by at most one, see [21, 22]. As an immediate consequence of one of our main results (Theorem 3), we have the following improvement to the Dol’nikov-Kříž lower bound.

Theorem 1

For any hypergraph $\mathcal{H}$ and any integer $r$ with $r\geq 2$, we have

[TABLE]

The following example illustrates the previous theorem is a true improvement to the Dol’nikov-Kříž lower bound. However, in the last section, we provide more examples in which the difference between the preceding lower bound and the Dol’nikov-Kříž lower bound can be arbitrary large.

Example 1

Let $n,k$ and $a$ be integers, where $n>a\geq rk-1$. Define $\mathcal{H}=\mathcal{H}(n,k,a)$ to be a hypergraph with vertex set $[n]$ and whose edge set is defined as follows:

[TABLE]

In the last section of this paper, we shall prove that $\chi\left(\operatorname{KG}^{r}(\mathcal{H})\right)=\left\lceil\dfrac{n-a}{r-1}\right\rceil.$ In fact, we will see that the preceding result simply follows from the lower bound stated in Theorem 1. What is interesting about Example 1 is that we cannot obtain the appropriate lower bound by using the colorability defect of $\mathcal{H}$ (the Dol’nikov-Kříž lower bound) or even the alternation number of $\mathcal{H}$. It should be noticed that, for any hypergraph $\mathcal{F}$, there is a lower bound for the chromatic number of $\operatorname{KG}^{r}(\mathcal{F})$ based on the alternation number of $\mathcal{F}$ which surpasses the Dol’nikov-Kříž lower bound (definitions and more details will be provided in Section 3.2). Some other examples comparing these lower bounds will be exhibited in Section 3.2.

Let $\mathcal{H}=\left([n],E(\mathcal{H})\right)$ be a hypergraph, $r$ an integer, and $\boldsymbol{s}=(s_{1},\ldots,s_{n})$ an integer vector, where $r\geq 2$ and $1\leq s_{i}<r$ for each $i\in[n]$. An $r$-(multi-)set $\left\{N_{1},\ldots,N_{r}\right\}$ is called $\boldsymbol{s}$-disjoint if each $i\in[n]$ appears in at most $s_{i}$ of the sets $N_{j}$ (we count the repetitions). Note that being $(1,\ldots,1)$-disjoint is the same thing as being pairwise disjoint. Ziegler [23] introduced the $\boldsymbol{s}$-disjoint general Kneser hypergraph $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})$ and the $\boldsymbol{s}$-disjoint $r$-colorability defect $\operatorname{cd}^{r}_{\boldsymbol{s}}(\mathcal{H})$ as generalizations of the general Kneser hypergraph $\operatorname{KG}^{r}(\mathcal{H})$ and the $r$-colorability defect $\operatorname{cd}^{r}(\mathcal{H})$, respectively. The $\boldsymbol{s}$*-disjoint general Kneser hypergraph $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})$ * is an $r$-uniform (multi-)hypergraph with vertex set $E(\mathcal{H})$ and edge set

[TABLE]

Depending on $\boldsymbol{s}$, an edge of $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})$ is not necessarily a set and it might be a multi-set of size $r$. Similar to the Kneser hypergraph $\operatorname{KG}^{r}(n,k)$, the $\boldsymbol{s}$-disjoint Kneser hypergraph $\operatorname{KG}^{r}_{\boldsymbol{s}}(n,k)$ is defined to be $\operatorname{KG}^{r}_{\boldsymbol{s}}\left([n],{[n]\choose k}\right)$. Also, the $\boldsymbol{s}$-disjoint $r$-colorability defect of $\mathcal{H}$, denoted by $\operatorname{cd}^{r}_{\boldsymbol{s}}(\mathcal{H})$, is defined as follows:

[TABLE]

where $\bar{n}=\bar{n}(\boldsymbol{s})=\sum\limits_{i=1}^{n}s_{i}$.

Ziegler [23, 24] generalized the Dol’nikov-Kříž lower bound to $\boldsymbol{s}$-disjoint general Kneser hypergraphs. Indeed, he proved that

[TABLE]

provided that $\max\limits_{i\in[n]}s_{i}<\mu(r)$, where $\mu(r)$ is the largest prime factor of $r$. Note that Ziegler’s lower bound immediately implies the Dol’nikov-Kříž lower bound since for $\boldsymbol{s}=(1,\ldots,1)$, we have $\operatorname{cd}^{r}_{\boldsymbol{s}}(\mathcal{H})=\operatorname{cd}^{r}(\mathcal{H})$ and $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})=\operatorname{KG}^{r}(\mathcal{H})$. Ziegler, using his lower bound, determined the chromatic number of the $\boldsymbol{s}$-disjoint general Kneser hypergraph $\operatorname{KG}^{r}_{\boldsymbol{s}}(n,k)$ for some integer vectors $\boldsymbol{s}$.

An $r$-uniform hypergraph $\mathcal{H}$ is called $r$-partite if its vertex set can be partitioned into subsets (parts) $V_{1},\ldots,V_{r}$ so that each of its edges intersects each part $V_{i}$ in exactly one vertex. An $r$-uniform $r$-partite hypergraph is said to be complete if it contains all possible edges. For the case $r=2$, such a graph is called a complete bipartite graph.

Generalizing Lovász’s result, several results concern the existence of large colorful complete bipartite subgraphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied, e.g. [2, 6, 7, 18, 19, 20]. In this regard, Simonyi and Tardos [20] proved that for an arbitrary hypergraph $\mathcal{H}$, any proper coloring of the general Kneser graph $\operatorname{KG}(\mathcal{H})$ contains a multicolored complete bipartite subgraph $K_{\lfloor{t\over 2}\rfloor,\lceil{t\over 2}\rceil}$ of order $t=\operatorname{cd}^{2}(\mathcal{H})$ in which the colors alternate on two parts of this bipartite subgraph with respect to their natural order. The existence of colorful subhypergraphs in hypergraphs coloring was first studied by Meunier [17]. Meunier generalized the Simonyi-Tardos result to general Kneser hypergraphs $\operatorname{KG}^{p}(\mathcal{H})$ provided that $p$ is prime. Actually, he proved that for a prime number $p$ and a hypergraph $\mathcal{H}$, any proper coloring of $\operatorname{KG}^{p}(\mathcal{H})$ contains a $p$-uniform $p$-partite subhypergraph with parts $V_{1},\ldots,V_{p}$ satisfying the following properties:

• •

$\displaystyle\sum_{i=1}^{p}|V_{i}|=\operatorname{cd}^{p}(\mathcal{H})$,

• •

the values of $|V_{i}|$’s differ by at most one, i.e., $||V_{i}|-|V_{j}||\leq 1$ for each $i,j\in[p]$, and

• •

for any $j\in[p]$, the vertices in $V_{j}$ get distinct colors.

The aforementioned result by Simonyi and Tardos ensures the existence of a colorful subgraph with $\operatorname{cd}^{2}(\mathcal{H})$ vertices in any proper coloring of $\operatorname{KG}(\mathcal{H})$. Some extensions of this result to more general topological settings can be found in [2, 18]. These results as well as Meunier’s were extended to the case of uniform hypergraphs in [1].

Let $r$ and $n$ be two integers and $\boldsymbol{s}=(s_{1},\ldots,s_{n})$ an integer vector, where $1\leq s_{i}<r$. For the sets $N_{1},\ldots,N_{r}\subseteq[n]$, the (multi-)set $\{N_{1},\ldots,N_{r}\}$ is called *equitable * if the values of $|N_{j}|$’s differ by at most one. Also, it is called equitable $\boldsymbol{s}$-disjoint if it is simultaneously equitable and $\boldsymbol{s}$-disjoint. For a hypergraph $\mathcal{H}=([n],E(\mathcal{H}))$, define the equitable $\boldsymbol{s}$-disjoint $r$-colorability defect of $\mathcal{H}$, denoted by $\operatorname{ecd}_{\boldsymbol{s}}^{r}(\mathcal{H})$, to be the following quantity:

[TABLE]

Note, here, $\bar{n}=\sum\limits_{i=1}^{n}s_{i}$. Throughout the paper, for $\boldsymbol{s}=(1,\ldots,1)$, since $\operatorname{ecd}^{r}(\mathcal{H})$ and $\operatorname{ecd}_{\boldsymbol{s}}^{r}(\mathcal{H})$ are the same, we use $\operatorname{ecd}^{r}(\mathcal{H})$ rather than $\operatorname{ecd}_{\boldsymbol{s}}^{r}(\mathcal{H})$.

The next theorem is the first main result of the paper which not only extends Meunier’s result [17] to general $\boldsymbol{s}$-disjoint Kneser hypergraphs $\operatorname{KG}^{p}_{\boldsymbol{s}}(\mathcal{H})$ but also improves it to equitable $r$-colorability defect. This theorem also results in a lower bound for the chromatic number of $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})$ based on the equitable $\boldsymbol{s}$-disjoint $r$-colorability defect $\operatorname{ecd}^{r}_{\boldsymbol{s}}(\mathcal{H})$ which surpasses Ziegler’s lower bound [23, 24].

Theorem 2

Let $\mathcal{H}=([n],E(\mathcal{H}))$ be a hypergraph, $p$ a prime number, and $\boldsymbol{s}=(s_{1},\ldots,s_{n})$ a positive integer vector, where $1\leq s_{i}<p$ for each $i\in[n]$. Any proper coloring of the $\boldsymbol{s}$-disjoint general Kneser hypergraph $\operatorname{KG}^{p}_{\boldsymbol{s}}(\mathcal{H})$ contains some subhypergraph whose vertex set can be partitioned into parts $V_{1},\ldots,V_{p}$ satisfying the following properties;

• •

$\sum\limits_{i=1}^{p}|V_{i}|=\operatorname{ecd}^{p}_{\boldsymbol{s}}(\mathcal{H})$,

• •

$\{e_{1},\ldots,e_{p}\}$* forms an edge of $\operatorname{KG}^{p}_{\boldsymbol{s}}(\mathcal{H})$ for each choice of $e_{i}\in V_{i}$,*

• •

the values of $|V_{i}|$’s differ by at most one, i.e., $||V_{i}|-|V_{j}||\leq 1$ for each $i,j\in[p]$, and

• •

for any $j\in[p]$, the vertices in $V_{j}$ get distinct colors.

If $\boldsymbol{s}=(1,\ldots,1)$, then Theorem 2 implies Meunier’s theorem in a stronger form (using $\operatorname{ecd}^{r}(\mathcal{H})$ instead of $\operatorname{cd}^{r}(\mathcal{H})$). Also, since any color appears in at most $p-1$ vertices of each edge of $\operatorname{KG}^{p}_{\boldsymbol{s}}(\mathcal{H})$, this theorem results in $\chi(\operatorname{KG}^{p}_{\boldsymbol{s}}(\mathcal{H}))\geq\left\lceil\dfrac{\operatorname{ecd}^{p}_{\boldsymbol{s}}(\mathcal{H})}{p-1}\right\rceil$ provided that $p$ is prime. The following theorem, applying this result, provides an improvement of Ziegler’s theorem [23, 24].

Theorem 3

Let $\mathcal{H}=([n],E(\mathcal{H}))$ be a hypergraph, $r$ an integer, $\boldsymbol{s}=(s_{1},\ldots,s_{n})$ a positive integer vector, and $\mu(r)$ be the largest prime factor of $r$. If $1\leq s_{i}<\mu(r)$ for each $i\in[n]$, then

[TABLE]

Since we always have $\operatorname{ecd}^{r}_{\boldsymbol{s}}(\mathcal{H})\geq\operatorname{cd}^{r}_{\boldsymbol{s}}(\mathcal{H})$, setting $\boldsymbol{s}=(1,\ldots,1)$ leads to the improvement of the Dol’nikov-Kříž lower bound stated in Theorem 1.

Plan. The rest of this paper is organized as follows. In Section 2, we first introduce some topological tools which will be needed elsewhere in the paper. Next, we prove Theorem 2 and deduce Theorem 3 from Theorem 2 by reducing Theorem 3 to the case of prime $r$. We conclude the paper in Section 3 by a discussion on comparing the equitable colorability defect with some other combinatorial parameters providing lower bounds for the chromatic number of general Kneser hypergraphs. Indeed, we will build some families of hypergraphs $\mathcal{H}$ in which the difference between the lower bound introduced in Theorem 3 and some other well-known lower bounds for the chromatic number of general Kneser hypergraphs $\operatorname{KG}^{r}(\mathcal{H})$ can be arbitrary large.

2 Proofs of the main results

This section is devoted to the proofs of Theorems 2 and 3. First, we review some basic definitions and tools. Next, we prove Theorem 2. To prove Theorem 3, we reduce this theorem to the case of prime $r$, which has been already proved by the discussion after Theorem 2.

2.1 Basic Tools

Here, a brief review of some notation and definitions which will be used throughout this section is provided though it is assumed that the reader has basic knowledge on topological combinatorics (for more, see [16]).

For an integer $r$, by $Z_{r}$, we refer to the cyclic multiplicative group of order $r$ and with generator $\omega$, i.e., $Z_{r}=\{\omega,\omega^{2},\ldots,\omega^{r}\}$. A simplicial complex, considered as a combinatorial object or a topological space, is a pair $(V,K)$, where $V$ - the vertex set - is a finite set and $K$ - the simplex set - is a hereditary system of nonempty subsets of $V$, that is, if $F\in K$ and $\varnothing\neq F^{\prime}\subseteq F$, then $F^{\prime}\in K$. By a simplicial complex $K$, we mean the simplicial complex $(V,K)$, where $V=\bigcup\limits_{F\in K}F$. Each set in $K$ is called a simplex of $K$. The dimension of $K$, denoted by $\dim(K)$, equals $\max\limits_{F\in K}|F|-1$. The first barycentric subdivision of the simplicial complex $K$, denoted by $\operatorname{sd}K$, is the order-complex obtained from the poset consisting of all simplices in $K$ ordered by inclusion. The join of two simplicial complexes $C$ and $K$, denoted by $C*K$, is a simplicial complex with vertex set $V(C)\uplus V(K)$ and simplex set

[TABLE]

In view of the definition, for three simplicial complexes $K,L$, and $M$, the two simplicial complexes $K*(L*M)$ and $(K*L)*M$ are the same simplicial complexes (up to a natural relabeling of their vertices). This allows us to use $K*L*M$ for both of $K*(L*M)$ and $(K*L)*M$ whenever we do not care about the names of the vertices. The join of $n$ disjoint copies of $K$ is denoted by $K^{*n}$. By renaming the vertices of $K^{*n}$, we can see $K^{*n}$ as a simplicial complex with vertex set $V(K)\times[n]$ such that for each vertex $(v,i)\in V(K)\times[n]$, the index $i$ indicates that $v$ is considered as a vertex of the $i$-th copy of $K$. Let $p$ be a prime and $s$ be an integer such that $1\leq s<p$. The simplicial complex $\sigma^{p-1}_{s-1}$ has $Z_{p}$ as the vertex set and its simplex set consists of all nonempty subsets of $Z_{p}$ with size at most $s$.

For an integer vector $\boldsymbol{s}=(s_{1},\ldots,s_{n})\in[p-1]^{n}$, the $\boldsymbol{s}$-disjoint $p$-fold join $(\sigma^{n-1})^{p}_{\boldsymbol{s}}$ is a simplicial complex whose vertex set is $Z_{p}\times[n]$ and its simplex set consists of all nonempty subsets $A$ of $Z_{p}\times[n]$ such that for each $i\in[n]$, the pair $(\varepsilon,i)$ is present in $A$ for at most $s_{i}$ different $\varepsilon\in Z_{p}$, i.e.,

[TABLE]

One can simply check that $(\sigma^{n-1})^{p}_{\boldsymbol{s}}=\sigma^{p-1}_{s_{1}-1}*\cdots*\sigma^{p-1}_{s_{n}-1}$.

Let $C$ and $K$ be two simplicial complexes. By a simplicial map $f:C\rightarrow K$, we mean a map from $V(C)$ to $V(K)$ such that the image of each simplex of $C$ is a simplex of $K$. The simplicial complex $C$ is a simplicial $Z_{p}$-complex if $Z_{p}$ acts on it and moreover, the map $\varepsilon:C\rightarrow C$ which maps $v$ to $\varepsilon\cdot v$ is a simplicial map for each $\varepsilon\in Z_{p}$. A simplicial $Z_{p}$-complex is called free if there is no fixed simplex under the simplicial map made by each $\varepsilon\in Z_{p}\setminus\{\boldsymbol{1}\}$, where $\boldsymbol{1}$ is the identity element of the group $Z_{p}$. For two simplicial $Z_{p}$-complexes $C$ and $K$, a simplicial map $f:C\rightarrow K$ is called $Z_{p}$-equivariant if $f(\varepsilon\cdot v)=\varepsilon\cdot f(v)$ for each $\varepsilon\in Z_{p}$ and $v\in V(C)$. Simply, any $Z_{p}$-equivariant simplicial map is called simplicial $Z_{p}$-map. Dold’s theorem [8] is a transformation group extension of the Borsuk-Ulam theorem which, in particular, asserts that if there is a simplicial $Z_{p}$-map from a free $Z_{p}$ space $C$ to a free $Z_{p}$ space $K$, then the dimension of $K$ is larger than the connectivity of $C$. For the definition of connectivity of topological spaces, we refer the reader to the book by Matoušek [16]. It is known that $\sigma^{p-1}_{s-1}$ is a free $(s-2)$-connected simplicial complex (with the natural $Z_{p}$-action). This along with the connectivity lemma for joins which asserts that the connectivity of the join of two topological spaces is at least the summation of their connectivity plus $2$ concludes that $(\sigma^{n-1})^{p}_{\boldsymbol{s}}$ (and its barycentric subdivision) has the connectivity $\bar{n}-2=\sum\limits_{i=1}^{n}s_{i}-2$ (see the topological proof of Lemma 5.3 in [23]).

2.2 Proof of Theorem 2

In our approach, we utilize the function $l(-)$ introduced by [1] and some sign functions proposed by Meunier [17]. Although, our technique is somehow similar to that used in [1], we will develop it to work here. To be more specific, the desired conclusion follows by applying Dold’s theorem to a simplicial $Z_{p}$-map defined via a given proper coloring of the general Kneser hypergraph between two simplicial $Z_{p}$-complexes. However, in [1], we did not care about the equitability and moreover we did not deal with multi-hypergraphs. Indeed, rather than in [1], we here work with different simplicial complexes, use sign functions differently, and moreover, we define the simplicial $Z_{p}$-map in a different way. Before starting the proof, we give some preliminaries. Let $q$ be a positive integer and $p$ a prime number. For a simplex $\tau\in\left(\sigma^{p-1}_{p-2}\right)^{*q}$, define $h(\tau)=\min\limits_{\varepsilon\in Z_{p}}|\tau^{\varepsilon}|,$ where \tau^{\varepsilon}=\big{\{}(\varepsilon,j):(\varepsilon,j)\in\tau\big{\}}. Also, set

[TABLE]

It should be emphasized that we shall use the two functions $l(-)$ and $h(-)$ several times during the proof.

**Proof **

Let $\mathcal{H}=([n],E(\mathcal{H}))$ be a hypergraph, $p$ a prime number, and $\boldsymbol{s}=(s_{1},\ldots,s_{n})$ a positive integer vector, where $1\leq s_{i}<p$ for each $i\in[n]$. Also, let $c:E(\mathcal{H})\longrightarrow[t]$ be a proper coloring of $\operatorname{KG}_{\boldsymbol{s}}^{p}(\mathcal{H})$. When $\operatorname{ecd}_{\boldsymbol{s}}^{p}(\mathcal{H})=0$, the statement of Theorem 2 is clearly true. For the rest of the proof, we shall assume that $\operatorname{ecd}_{\boldsymbol{s}}^{p}(\mathcal{H})>0$.

Hereafter, for simplicity of notation, we set $K=\sigma_{s_{1}-1}^{p-1}*\cdots*\sigma_{s_{n}-1}^{p-1}$. For a positive integer $a\in[n]$, let $W_{a}$ denote the set consisting of all simplices $\tau\in K$ such that $|\tau^{\varepsilon}|\in\{0,a\}$ for each $\varepsilon\in Z_{p}$. Also, for a positive integer $b\in[t]$, let $U_{b}$ be the set consisting of all simplices $\tau\in\left(\sigma^{p-1}_{p-2}\right)^{*t}$ such that $|\tau^{\varepsilon}|\in\{0,b\}$ for each $\varepsilon\in Z_{p}$. Define $W=\bigcup\limits_{a=1}^{n}W_{a}$ and $U=\bigcup\limits_{b=1}^{t}U_{b}$. Choose three arbitrary $Z_{p}$-equivariant maps $s_{0}:\sigma^{p-1}_{p-2}\rightarrow Z_{p}$, $s_{1}:W\rightarrow Z_{p}$, and $s_{2}:U\rightarrow Z_{p}$ by choosing one representative in each orbit (note that this is possible because $Z_{p}$ acts freely on each of $\sigma^{p-1}_{p-2}$, $W$, and $U$).

For each simplex $\tau$ of $K$ and for each $\varepsilon\in Z_{p}$, define

[TABLE]

Note that $\tau_{\varepsilon}\subseteq[n]=V(\mathcal{H}).$ Further, define

[TABLE]

When $\{\tau\in K\colon E(\mathcal{H}[\tau_{\varepsilon}])=\varnothing\mbox{ for each }\varepsilon\in Z_{p}\}=\varnothing$, i.e., each singleton is an edge of $\mathcal{H}$, we define $l(\mathcal{H})=0$. One should notice that $\bar{n}-l(\mathcal{H})=\operatorname{ecd}_{\boldsymbol{s}}^{r}(\mathcal{H})$. Now, set

[TABLE]

where, for each simplex $\tau$ in $K$,

[TABLE]

Since $\operatorname{ecd}_{\boldsymbol{s}}^{r}(\mathcal{H})=\bar{n}-l(\mathcal{H})>0$, the set \{l(\tau_{c})\colon\tau\in K\mbox{ and }l(\tau)>l(\mathcal{H})\big{\}} is not empty and consequently $m$ is well defined. In view of the definition of $K=(\sigma^{n-1})^{p}_{\boldsymbol{s}}$, for each simplex $\tau$ of $K$, the set $\{\tau_{\varepsilon}\colon\varepsilon\in Z_{p}\}$ is $\boldsymbol{s}$-disjoint. This observation along with the fact that $c$ is a proper coloring of $\operatorname{KG}^{p}_{\boldsymbol{s}}(\mathcal{H})$ implies $\tau_{c}$ is either empty or a simplex of $(\sigma_{p-2}^{p-1})^{*t}$. Indeed, if $l(\tau)>l(\mathcal{H})$, then $\tau_{c}$ is a simplex of $(\sigma_{p-2}^{p-1})^{*t}$. This observation plays a crucial role in our proof.

In what follows, we first define a map $\gamma\colon\operatorname{sd}K\longrightarrow Z_{p}^{*m}.$ Let $\tau$ be an arbitrary simplex in $K$. Define $\gamma(\tau)$ as follows.

• a)

For $l(\tau)\leq l(\mathcal{H})$, we consider the two following cases.

If $h(\tau)=0$, then define $\gamma(\tau)=(s_{0}(\bar{\tau}),l(\tau))$, where

[TABLE] 2. 2.

If $h(\tau)>0$, then define $\gamma(\tau)=\left(s_{1}(\bar{\tau}),l(\tau)\right)$, where

[TABLE]

• b)

For $l(\tau)>l(\mathcal{H})$, the set $\tau_{c}$ is not empty and thus it is a simplex of $(\sigma_{p-2}^{p-1})^{*t}$. Now, we consider two different cases.

If $h(\tau_{c})=0$, then define $\gamma(\tau)=(s_{0}(\bar{\tau}_{c}),l(\mathcal{H})+l(\tau_{c}))$, where

[TABLE] 2. 2.

If $h(\tau_{c})>0$, then define $\gamma(\tau)=\left(s_{2}(\bar{\tau}_{c}),l(\mathcal{H})+l(\tau_{c})\right)$, where

[TABLE]

Claim 1

The map $\gamma$ is a simplicial $Z_{p}$-map from $\operatorname{sd}K$ to $Z_{p}^{*m}$.

**Proof **

First, note that all the functions $s_{0}(-)$, $s_{1}(-)$, and $s_{2}(-)$ are $Z_{p}$-equivariant maps. Accordingly, it is clear that $\gamma$ is a $Z_{p}$-equivariant map as well. For a contradiction, suppose that $\gamma$ is not a simplicial map. In view of the simplices in $\operatorname{sd}K$ and $Z_{p}^{*m}$, we conclude there are two simplices $\tau$ and $\tau^{\prime}$ in $K$ such that $\tau\subsetneq\tau^{\prime}$, $\gamma(\tau)=(\varepsilon_{1},\beta)$, and $\gamma(\tau^{\prime})=(\varepsilon_{2},\beta)$, where $\varepsilon_{1}\neq\varepsilon_{2}$ and $\beta\in[m]$. In view of the definition of $\gamma$, it is clear that it is not possible to have $l(\tau)\leq l(\mathcal{H})$ and $l(\tau^{\prime})>l(\mathcal{H})$, simultaneously. Therefore, we have either $l(\tau)\leq l(\tau^{\prime})\leq l(\mathcal{H})$ or $l(\tau^{\prime})\geq l(\tau)>l(\mathcal{H})$, which will be discussed separately in what follows.

• I)

$l(\tau)\leq l(\tau^{\prime})\leq l(\mathcal{H}).$ Clearly, in view of the definition of $\gamma$ in this case, we should have $l(\tau)=l(\tau^{\prime})$. We consider the two following cases.

If $h(\tau)=h(\tau^{\prime})=0$, then since $\tau\subsetneq\tau^{\prime}$ and

[TABLE]

we conclude $\{\varepsilon\in Z_{p}\colon{\tau^{\prime}}^{\varepsilon}=\varnothing\}\subsetneq\{\varepsilon\in Z_{p}\colon\tau^{\varepsilon}=\varnothing\}$. This implies that

[TABLE]

a contradiction. 2. 2.

For $h(\tau^{\prime})>0$, we know that

[TABLE]

The facts $l(\tau)=l(\tau^{\prime})$ and

[TABLE]

imply that $h=h(\tau)=h(\tau^{\prime})$ and

[TABLE]

In view of the equations

[TABLE]

we must have

[TABLE]

This inequality in combination with the facts that $\tau\subsetneq\tau^{\prime}$ and $h=\displaystyle\min_{\varepsilon\in Z_{p}}|\tau^{\varepsilon}|=\displaystyle\min_{\varepsilon\in Z_{p}}|{\tau^{\prime}}^{\varepsilon}|$ imply that

[TABLE]

which is impossible.

• II)

$l(\tau^{\prime})\geq l(\tau)>l(\mathcal{H}).$ Note both $\tau_{c}$ and ${\tau^{\prime}}_{c}$ are simplices in $(\sigma_{p-2}^{p-1})^{*t}$. Clearly, in view of the definition of $\gamma$ in this case, we should have $l(\tau_{c})=l({\tau^{\prime}}_{c})$. Also, note that since $\tau\subsetneq\tau^{\prime}$, we have $\tau_{c}\subseteq{\tau^{\prime}}_{c}$. Furthermore, the equality cannot happen, since otherwise, in view of the definition of $\gamma$, we must have $\varepsilon_{1}=\varepsilon_{2}$, which contradicts the way $\tau$ and $\tau^{\prime}$ have been chosen. Similar to the previous case, we will deal with the two different cases $h(\tau_{c})=h({\tau^{\prime}}_{c})=0$ and $h({\tau^{\prime}}_{c})>0$.

If $h(\tau_{c})=h({\tau^{\prime}}_{c})=0$, then since $\tau_{c}\subsetneq\tau^{\prime}_{c}$ and

[TABLE]

we have $\{\varepsilon\in Z_{p}\colon{\tau^{\prime}}_{c}^{\varepsilon}=\varnothing\}\subsetneq\{\varepsilon\in Z_{p}\colon\tau_{c}^{\varepsilon}=\varnothing\}$. This implies that

[TABLE]

a contradiction. 2. 2.

For $h({\tau^{\prime}}_{c})>0$, we know that

[TABLE]

The facts $l(\tau_{c})=l({\tau^{\prime}}_{c})$ and

[TABLE]

conclude $h=h(\tau_{c})=h({\tau^{\prime}}_{c})$ and

[TABLE]

According to

[TABLE]

we must have

[TABLE]

This inequality and the facts that $\tau_{c}\subsetneq{\tau^{\prime}}_{c}$ and $h=\displaystyle\min_{\varepsilon\in Z_{p}}|\tau_{c}^{\varepsilon}|=\displaystyle\min_{\varepsilon\in Z_{p}}|{\tau^{\prime}}_{c}^{\varepsilon}|$ result in

[TABLE]

which is impossible.

$\blacksquare$

Claim 2

There is a simplex $\tau\in K$ for which $l({\tau}_{c})\geq\operatorname{ecd}_{\boldsymbol{s}}^{p}(\mathcal{H}).$

**Proof **

By Claim 1, it has already been noted that $\gamma$ is a simplicial $Z_{p}$-map from $\operatorname{sd}K$ to $Z_{p}^{*m},$ where

[TABLE]

Accordingly, in view of Dold’s theorem [8], the dimension of $Z_{p}^{*m}$ must be strictly larger than the connectivity of $\operatorname{sd}K$; that is $m-1>\bar{n}-2$, which implies $m\geq\bar{n}$ (see the discussion at the end of Subsection 2.1). Consequently, from the definition of $m$, there is a simplex $\tau\in K$ for which $l(\mathcal{H})+l({\tau}_{c})\geq\bar{n}.$ Equivalently, we have

[TABLE]

as desired.

• $\blacksquare$*

Let $\tau$ be a simplex for which $l({\tau}_{c})\geq\operatorname{ecd}_{\boldsymbol{s}}^{p}(\mathcal{H})$. For simplicity of notation, set $h=h(\tau_{c})$. For each $i\in[p]$, if $|{\tau_{c}}^{\omega^{i}}|=h$, then define $C_{i}$ to be the $h$-set $\{c_{i,1},\ldots,c_{i,h}\}\subseteq[t]$ for which $\{\omega^{i}\}\times C_{i}={\tau_{c}}^{\omega^{i}}$, otherwise, (if $|{\tau_{c}}^{\omega^{i}}|>h$) let $C_{i}=\{c_{i,1},\ldots,c_{i,h+1}\}\subseteq[t]$ be an $(h+1)$-set such that $\{\omega^{i}\}\times C_{i}\subseteq{\tau_{c}}^{\omega^{i}}$. Now, for each $i\in[p]$, if $|{\tau_{c}}^{\omega^{i}}|=h$, then define $V_{i}=\left\{e_{i,1},\ldots,e_{i,h}\right\}$ such that $e_{i,j}\subseteq\tau_{\omega^{i}}$ and $c(e_{i,j})=c_{i,j}$ for each $j\in[h]$, otherwise, define $V_{i}=\left\{e_{i,1},\ldots,e_{i,h+1}\right\}$ such that $e_{i,j}\subseteq\tau_{\omega^{i}}$ and $c(e_{i,j})=c_{i,j}$ for each $j\in[h+1]$. Clearly,

[TABLE]

We have already noticed that the set $\{\tau_{\varepsilon}\colon\varepsilon\in Z_{p}\}$ is $\boldsymbol{s}$-disjoint, therefore, for each choice of $e_{i}\in V_{i}$, the set $\{e_{1},\ldots,e_{p}\}$ is $\boldsymbol{s}$-disjoint as well and consequently $\{e_{1},\ldots,e_{p}\}$ is an edge of $\operatorname{KG}^{p}_{\boldsymbol{s}}(\mathcal{H})$. It is then trivial that the subhypergraph $\operatorname{KG}^{p}_{\boldsymbol{s}}(\mathcal{H})[\bigcup\limits_{i=1}^{p}V_{i}]$ is the desired subhypergraph.

• $\blacksquare$*

2.3 Proof of Theorem 3

In this subsection, we reduce Theorem 3 to the case of prime $r$, which is known to be true owing to Theorem 2 (see the discussion after Theorem 2). The idea of this reduction is originally due to Kříž [14], which has been used in other papers as well, e.g. [2, 11, 23, 24].

For a hypergraph $\mathcal{H}=([n],E(\mathcal{H}))$ and positive integers $r$ and $C$, define $\mathcal{T}_{\mathcal{H},C,r}$ to be a hypergraph with vertex set $[n]$ and edge set

[TABLE]

Lemma 4

Let $r^{\prime}$, $r^{\prime\prime}$ be two positive integers and $\boldsymbol{s}=(s_{1},\ldots,s_{n})$ be an integer vector, where $1\leq s_{i}<r^{\prime\prime}$. For any hypergraph $\mathcal{H}=([n],E(\mathcal{H}))$, the following inequality holds

[TABLE]

**Proof **

For the ease of use, set $\mathcal{T}=\mathcal{T}_{\mathcal{H},C,r^{\prime}}$. In view of the definition of $\operatorname{ecd}^{r^{\prime\prime}}_{\boldsymbol{s}}(\mathcal{T})$, there exists a family of equitable $\boldsymbol{s}$-disjoint subsets of $[n]$, say $\{V_{1},...,V_{r^{\prime\prime}}\}$, such that $\sum\limits_{i=1}^{r^{\prime\prime}}|V_{i}|=\sum\limits_{i=1}^{n}s_{i}-\operatorname{ecd}^{r^{\prime\prime}}_{\boldsymbol{s}}(\mathcal{T})$ and $E(\mathcal{T}[V_{i}])=\varnothing$ for each $i\in[r^{\prime\prime}]$. Clearly, this implies that $V_{i}\not\in E(\mathcal{T})$ for each $i\in[r^{\prime\prime}]$ and consequently, $\operatorname{ecd}^{r^{\prime}}(\mathcal{H}[V_{i}])\leq(r^{\prime}-1)C$ for each $i\in[r^{\prime\prime}]$. Thus, for each $i\in[r^{\prime\prime}]$, there is a family $\{V_{i1},...,V_{ir^{\prime}}\}$ of equitable disjoint subsets of $V_{i}$ such that

[TABLE]

and $E(\mathcal{H}[V_{ij}])=\varnothing$ for all $j\in[r^{\prime}]$. One can simply check that the family $\left\{V_{ij}\colon 1\leq i\leq r^{\prime\prime}\;\&\;1\leq j\leq r^{\prime}\right\}$ is an equitable $\boldsymbol{s}$-disjoint family of subsets of $[n]$ and moreover, $E(\mathcal{H}[V_{ij}])=\varnothing$ for each $i\in[r^{\prime\prime}]$ and $j\in[r^{\prime}]$. Accordingly,

[TABLE]

which completes the proof.

• $\blacksquare$*

Lemma 5

Let $r,r^{\prime},r^{\prime\prime}$ be positive integers, where $r^{\prime},r^{\prime\prime}\geq 2$, $r=r^{\prime}r^{\prime\prime}$. Also, let $\boldsymbol{s}=(s_{1},\ldots,s_{n})$ be a positive integer vector, where $1\leq s_{i}<r^{\prime\prime}$ for each $i\in[r^{\prime\prime}]$. If Theorem 3 holds for $r^{\prime}$ and $\boldsymbol{s}^{\prime}=(1,\ldots,1)$ and also for $r^{\prime\prime}$ and $\boldsymbol{s}$, then it holds for $r$ and $\boldsymbol{s}$.

**Proof **

For contradiction sake, suppose that there is a proper coloring $c:E(\mathcal{H})\rightarrow[C]$ of $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})$ for which $\operatorname{ecd}^{r}_{\boldsymbol{s}}(\mathcal{H})>(r-1)C$. Applying Lemma 4 leads us to

[TABLE]

which immediately implies $(r^{\prime\prime}-1)C<\operatorname{ecd}^{r^{\prime\prime}}_{\boldsymbol{s}}(\mathcal{T}_{\mathcal{H},C,r^{\prime}}).$ Since Theorem 3 holds for $r^{\prime\prime}$ and $\boldsymbol{s}$, the preceding observation concludes

[TABLE]

On the other hand, in view of the definition of $\mathcal{T}_{{\mathcal{H},C,r^{\prime}}}$, for each $e\in E(\mathcal{T}_{\mathcal{H},C,r^{\prime}})$, we have $\operatorname{ecd}^{r^{\prime}}(\mathcal{H}[e])>(r^{\prime}-1)C$. Since Theorem 3 holds for $r^{\prime}$ and $\boldsymbol{s}^{\prime}=(1,\ldots,1)$, this implies $\chi(\operatorname{KG}^{r^{\prime}}(\mathcal{H}[e]))>C$. Consequently the assignment of colors to the vertices of $\operatorname{KG}^{r^{\prime}}(\mathcal{H}[e])$ through the coloring $c$ cannot be a proper coloring. Therefore, for each edge $e\in E(\mathcal{T}_{\mathcal{H},C,r^{\prime}})$, there is at least one monochromatic edge in $\operatorname{KG}^{r^{\prime}}(\mathcal{H}[e])$. Now, for each edge $e\in E(\mathcal{T}_{\mathcal{H},C,r^{\prime}})$, set $h(e)$ to be the largest color amongst the colors of monochromatic edges in $\operatorname{KG}^{r^{\prime}}(\mathcal{H}[e])$. Since $c$ is a proper coloring of $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})$, the map $h:E(\mathcal{H})\rightarrow C$ is a proper coloring of $\operatorname{KG}^{r^{\prime\prime}}_{\boldsymbol{s}}(\mathcal{T}_{\mathcal{H},C,r^{\prime}})$, which implies $\chi(\operatorname{KG}^{r^{\prime\prime}}_{\boldsymbol{s}}(\mathcal{T}_{\mathcal{H},C,r^{\prime}}))\leq C,$ a contradiction. To see this, contrary to the claim, suppose that $\{f_{1},\ldots,f_{r^{\prime\prime}}\}$ is an edge of $\operatorname{KG}^{r^{\prime\prime}}_{\boldsymbol{s}}(\mathcal{T}_{\mathcal{H},C,r^{\prime}})$ such that $h(f_{1})=\cdots=h(f_{r^{\prime\prime}})=a$. By the definition of $h(-)$, for each $j\in[r^{\prime\prime}]$, there is a monochromatic edge $\{e_{1,j},\ldots,e_{r^{\prime},j}\}$ of $\operatorname{KG}^{r^{\prime}}(\mathcal{H}[f_{j}])$ for which we have $c(e_{1,j})=\cdots=c(e_{r^{\prime},j})=a$. One can simply see that $\left\{e_{i,j}\colon i\in[r^{\prime}]\;\&\;j\in[r^{\prime\prime}]\right\}$ is a monochromatic edge of $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})$, contradicting the fact that $c$ is a proper coloring for $\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H})$.

• $\blacksquare$*

Here, we apply Lemma 5 to complete the proof of Theorem 3. First, note that, by the discussion after Theorem 2, we know that Theorem 3 is true for $\boldsymbol{s}=(1,\ldots,1)$ and the prime values of $r$. Thus, applying Lemma 5 with $\boldsymbol{s}=\boldsymbol{s}^{\prime}=(1,\ldots,1)$, Theorem 3 holds for $\boldsymbol{s}=(1,\ldots,1)$ and any positive integer $r\geq 2$. Now, in Lemma 5, we set $r^{\prime\prime}$ to be the largest prime factor of $r$. Again, in view of the discussion after Theorem 2, we know that Theorem 3 holds for $r^{\prime\prime}$ and $\boldsymbol{s}$. Consequently, applying Lemma 5 gives the desired conclusion.

3 Comparing equitable colorability defect with colorability defect and alternation number

In this section, we compare equitable colorability defect of hypergraphs with colorability defect and alternation number of them, two other combinatorial parameters providing lower bounds for the chromatic number of general Kneser hypergraphs.

3.1 Comparing equitable colorability defect with colorability defect

By the definitions of $\operatorname{cd}_{\boldsymbol{s}}^{r}(\mathcal{H})$ and $\operatorname{ecd}_{\boldsymbol{s}}^{r}(\mathcal{H})$, it is apparent $\operatorname{ecd}_{\boldsymbol{s}}^{r}(\mathcal{H})\geq\operatorname{cd}_{\boldsymbol{s}}^{r}(\mathcal{H})$. There are several examples ensuring that not only this inequality might be strict but also the difference between $\operatorname{ecd}_{\boldsymbol{s}}^{r}(\mathcal{H})$ and $\operatorname{cd}_{\boldsymbol{s}}^{r}(\mathcal{H})$ can be arbitrary large. To see this, let $m$ and $n$ be positive integers, where $m\geq r(n+1)$. Consider an arbitrary hypergraph $\mathcal{F}$ with $n$ vertices and let $U=\{u_{1},\ldots,u_{m}\}$ be a set disjoint from $V(\mathcal{F})$. Define $\mathcal{H}$ to be a hypergraph with vertex set $V(\mathcal{F})\cup U$ and edge set $E(\mathcal{H})=E(\mathcal{F})\cup\left\{\{u,v\}\colon u\in U\mbox{ and }v\in V(\mathcal{F})\right\}.$ Since any independent set of $\mathcal{H}$ is either a subset of $U$ or an independent set of $\mathcal{F}$, it is simple to check that $\operatorname{cd}^{r}(\mathcal{H})=\operatorname{cd}^{r-1}(\mathcal{F})$ and $\operatorname{ecd}^{r}(\mathcal{H})=n$, which results in $\operatorname{ecd}^{r}(\mathcal{H})-\operatorname{cd}^{r}(\mathcal{H})=n-\operatorname{cd}^{r-1}(\mathcal{F})$. Now, one can consider several hypergraphs $\mathcal{F}$ for which $n-\operatorname{cd}^{r-1}(\mathcal{F})$ is arbitrary large. By the preceding construction, in particular, we can build some $r$-partite graphs which are also discussed in more detail in the following proposition.

Proposition 6

If we set $\mathcal{H}$ to be the complete $r$-partite graph $K_{t,...t,T}$ with $T\geq rt+r-1$, then $\operatorname{cd}^{r}(\mathcal{H})=0$ while $\chi(\operatorname{KG}^{r}(\mathcal{H}))={\operatorname{ecd}^{r}(\mathcal{H})\over r-1}=t$.

**Proof **

Let $V(\mathcal{H})=V_{1}\uplus\cdots\uplus V_{r}$ such that each $V_{i}$ is independent, $|V_{1}|=\cdots=|V_{r-1}|=t$, and $|V_{r}|=T$. First note that $\mathcal{H}$ is $r$-colorable (it is $r$-partite) which clearly implies $\operatorname{cd}^{r}(\mathcal{H})=0$. Let $(S_{1},\ldots,S_{r})$ be an equitable $r$-coloring of \mathcal{H}\big{[}\bigcup\limits_{i=1}^{r}S_{i}\big{]}. To prove $\operatorname{ecd}^{r}(\mathcal{H})=(r-1)t$, we need to show that $|\bigcup\limits_{i=1}^{r}S_{i}|\leq T$ and the equality can be achieved, which will be concluded from the following two cases: If there is some $i\in[r]$ such that $S_{i}\subseteq V_{j}$ and $j\leq r-1$, then since $(S_{1},\ldots,S_{r})$ is an equitable partition for $\bigcup\limits_{i=1}^{r}S_{i}$, we have $|\bigcup\limits_{i=1}^{r}S_{i}|\leq t+(r-1)(t+1)\leq T$. Otherwise, if $S_{i}\subseteq V_{r}$ for each $i\in[r]$, then $\bigcup\limits_{i=1}^{r}S_{i}\subseteq V_{r}$ which implies $|\bigcup\limits_{i=1}^{r}S_{i}|\leq|V_{r}|=T$. It is clear that we can have the equality in latter case. In view of Theorem 1, to finish the proof, it suffices to show that $\operatorname{KG}^{r}(\mathcal{H})$ is $t$-colorable. To this end, let $V(\mathcal{H})\setminus V_{r}=U_{1}\uplus\cdots\uplus U_{t}$ such that $|U_{j}|=r-1$ for each $j\in[t]$. Now, for each edge $e\in E(\mathcal{H})$, define $c(e)$ to be the minimum $j$ such that $e\cap U_{j}\neq\varnothing.$ One can check that $c$ is a proper $t$-coloring of $\operatorname{KG}^{r}(\mathcal{H})$, as desired.

• $\blacksquare$*

Let us remind that $\mu(r)$ is the largest prime factor of an integer $r\geq 2$. Ziegler [23, 24], generalizing the Alon-Frankl-Lovász theorem [4], determined the chromatic number of some families of $\boldsymbol{s}$-disjoint Kneser hypergraphs $\operatorname{KG}_{\boldsymbol{s}}^{r}(n,k)$. For positive integers $n,k,r,$ and $s$ with $k\geq 2$, $ns\geq rk$, and $1\leq s<\mu(r)$, he proved

[TABLE]

provided that $s$ divides $r-1$. Note that if $s=1$, then we have the Alon-Frankl-Lovász theorem.

Let $n,k,r,$ and $a$ be positive integers and $\boldsymbol{s}\in[r-1]^{n}$ a positive integer vector with $k,r\geq 2$, $n>a$, and $\sum\limits_{i=1}^{n}s_{i}\geq rk$. Define $\mathcal{H}(n,k,a)$ to be a hypergraph with vertex set $[n]$ and edge set $\left\{e\in{[n]\choose k}:e\nsubseteq\{n-a+1,\ldots,n\}\right\}$. Also, set $\operatorname{KG}^{r}_{\boldsymbol{s}}(n,k,a)=\operatorname{KG}^{r}_{\boldsymbol{s}}(\mathcal{H}(n,k,a))$. Note that $\operatorname{KG}^{r}_{\boldsymbol{s}}(n,k,a)=\operatorname{KG}^{r}_{\boldsymbol{s}}(n,k)$ when $a\leq k-1$. In what follows, we compare the equitable colorability defect and colorability defect of $\mathcal{H}(n,k,a)$ and prove that the difference between these two quantities can be arbitrary large. Moreover, extending the latter result by Ziegler, we study the chromatic number of $\operatorname{KG}_{(s,\ldots,s)}^{r}(n,k,a)$ in some cases.

Lemma 7

Let $n,k,r,$ and $a$ be positive integers and $\boldsymbol{s}=(s,\ldots,s)\in[r-1]^{n}$ a positive integer vector with $k,r\geq 2$, $n>a$, and $ns\geq rk$. Then, we have

[TABLE]

**Proof **

The proof is similar to that of Lemma 3.1 in [23]. Set $P=\lfloor{r-1\over s}\rfloor$ and $M=\left\lceil{1\over\lfloor{r-1\over s}\rfloor}{ns-\max\{rk-1,sa+r-1\}\over s}\right\rceil$. For each vertex $e\in{[n]\choose k}$, define

[TABLE]

Note that $c$ is a coloring with color set $[M+1]$. To complete the proof, it suffices to show that this coloring is proper. Contradictory, suppose that $\{e_{1},\ldots,e_{r}\}$ is an edge of $\operatorname{KG}^{r}_{\boldsymbol{s}}(n,k,a)$ with $c(e_{1})=\cdots=c(e_{r})=\ell$. First note that $\ell\leq M$ is impossible. Indeed, since $Ps=\lfloor{r-1\over s}\rfloor s\leq r-1$, at most $r-1$ of the edges $e_{i}$’s can receive a color $\ell\in[M]$. If $\ell=M+1$, then $\min e_{i}>MP$ for each $i\in[r]$ which concludes $\bigcup\limits_{i=1}^{r}e_{i}\subseteq\{MP+1,\ldots,n\}$. We thus have

[TABLE]

which implies $sa+r-1\geq rk$ and consequently $MP+1\geq n-a-\lfloor{r-1\over s}\rfloor+1$. Since $\operatorname{KG}_{\boldsymbol{s}}^{r}(n,k,a)$ has no vertex contained in $\{n-a+1,\ldots,n\}$, each $e_{i}$ must have some element in $\left\{n-a-\lfloor{r-1\over s}\rfloor+1,\ldots,n-a\right\}$. On the other hand, since $\{e_{1},\ldots,e_{r}\}$ is an $(s,\ldots,s)$-disjoint family, each $j\in[n]$ is present in at most $s$ of the sets $e_{i}$ which concludes $s\lfloor{r-1\over s}\rfloor\geq r$, a contradiction.

• $\blacksquare$*

By the following observation, we will compute the $\boldsymbol{s}$-disjoint $r$-colorability defect and the equitable $\boldsymbol{s}$-disjoint $r$-colorability defect of $\mathcal{H}(n,k,a)$ when $\boldsymbol{s}=(s,\ldots,s)\in[r-1]^{n}$.

Observation 8

Let $n,k,r,$ and $a$ be positive integers and $\boldsymbol{s}=(s,\ldots,s)\in[r-1]^{n}$ a positive integer vector with $k,r\geq 2$, $n>a$, and $ns\geq rk$. Then

[TABLE]

and

[TABLE]

**Sketch of Proof **

The proof of Observation 8 is simple but technical. To ease the reading, we just sketch the proof of the second equality. Let $\{N_{1},\ldots,N_{r}\}$ be an equitable $(s,\ldots,s)$-disjoint family of subsets of $[n]$ such that $\operatorname{ecd}^{r}_{\boldsymbol{s}}(\mathcal{H}(n,k,a))=ns-\sum\limits_{i=1}^{r}|N_{i}|$ and \mathcal{H}(n,k,a)\big{[}N_{j}\big{]}=\varnothing for each $j\in[r]$. First note that $\min\limits_{j\in[r]}|N_{j}|\geq k-1$ and also, for each $N_{j}$ with $|N_{j}|\geq k$, we must have $N_{j}\subseteq A=\{n-a+1,\ldots,n\}$. Consequently, one can check that $\sum\limits_{j=1}^{r}|N_{j}|$ will be maximized whenever the number of $N_{j}$’s with $N_{j}\subseteq A$ and $|N_{j}|\geq k$ is maximized. Now, one can simply verify the desired equations.

• $\blacksquare$*

By the next corollary, generalizing Ziegler’s result [23, 24], we compute the chromatic number of $\operatorname{KG}^{r}_{\boldsymbol{s}}(n,k,a)$ in some cases.

Corollary 9

Let $n,k,r,$ and $a$ be positive integers and $\boldsymbol{s}=(s,\ldots,s)\in[\mu(r)-1]^{n}$ be a positive integer vector with $k,r\geq 2$, $n>a$, and $ns\geq rk$. If $s$ divides $r-1$, then

[TABLE]

provided that either $a\leq k-1$ or $as\geq rk-1$.

**Proof **

Since $s$ divides $r-1$, by Theorem 3 and Lemma 7, we have

[TABLE]

Now, the proof follows by Observation 8.

• $\blacksquare$*

Under the same assumption as in Corollary 9, if $s$ divides $r-1$ and $ks\leq as<rk-1$, then, by Lemma 7, Theorem 3, and Observation 8, we have

[TABLE]

Therefore, using Theorem 3, we are not able to determine the chromatic number of $\operatorname{KG}_{\boldsymbol{s}}^{r}(n,k,a)$ in this case. As an interesting question, one may ask for the chromatic number of $\operatorname{KG}_{\boldsymbol{s}}^{r}(n,k,a)$ provided that $ks\leq as<rk-1$ and $s$ divides $r-1$. In Subsection 3.2, we present more evidences supporting the supposition that the chromatic number of $\operatorname{KG}_{\boldsymbol{s}}^{r}(n,k,a)$ is equal to the preceding upper bound provided that $ks\leq as<rk-1$ and $s$ divides $r-1$.

3.2 Comparing equitable colorability defect with alternation number

Throughout this subsection, assume $\boldsymbol{s}=(1,\ldots,1)$. Here, we first define the alternation number of hypergraphs. For an $X=(x_{1},...,x_{n})\in(Z_{r}\cup\{0\})^{n}$, the subsequence $x_{i_{1}},\ldots,x_{i_{m}}$ of nonzero coordinates of $X$ is called alternating if any two consecutive terms of this subsequence are different. In other words, the sequence $x_{i_{1}},\ldots,x_{i_{m}}$ is called alternating whenever $1\leq i_{1}<\cdots<i_{m}\leq n$, $x_{i_{j}}\neq 0$ for each $j\in[m]$ and $x_{i_{j}}\neq x_{i_{j+1}}$ for each $j\in[m-1]$. For each $X\in(Z_{r}\cup\{0\})^{n}$, we define $\operatorname{alt}(X)$ to be the longest alternating subsequence of $X$. Also, we define $\operatorname{alt}(\boldsymbol{0})=0$. For each $X\in(Z_{r}\cup\{0\})^{n}$ and for each $i\in[r]$, set $X^{i}=\{j\in[n]\colon x_{j}=\omega^{i}\}.$ By abuse of notation, we can write $X=(X^{1},\ldots,X^{r})$.

Let $\mathcal{H}=(V(\mathcal{H}),E(\mathcal{H}))$ be a hypergraph. Also, consider a bijection $\sigma:[n]\longrightarrow V(\mathcal{H})$. The $r$-alternation number of $\mathcal{H}$ with respect to the bijection $\sigma$, denoted by $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})$, is the maximum possible $k$ for which there is an $X\in(Z_{r}\cup\{0\})^{n}$ with $\operatorname{alt}(X)=k$ such that $E(\mathcal{H}[\sigma(X^{i})])=\varnothing$ for each $i\in[r]$, i.e.,

[TABLE]

Define

[TABLE]

where the minimum is taken over all bijections $\sigma:[n]\longrightarrow V(\mathcal{H})$. The second present author and Hajiabolhassan [2, Theorem 3], using the $Z_{p}$-Tucker lemma, proved that

[TABLE]

They also computed the chromatic number of several families of hypergraphs using this lower bound, see [2]. One can simply see that the preceding lower bound surpasses the Dol’nikov-Kříž lower bound.

Note that, setting $\boldsymbol{s}=(1,\ldots,1)$, we have already computed $\operatorname{cd}^{r}(\mathcal{H}(n,k,a))$ and $\operatorname{ecd}^{r}(\mathcal{H}(n,k,a))$ by Observation 8. However, for computing $\operatorname{alt}^{r}(\mathcal{H}(n,k,a))$, we need to put more effort.

Lemma 10

Let $n,k,r,$ and $a$ be positive integers such that $k,r\geq 2$ and $n\geq\max\{a+1,rk\}$. Then

[TABLE]

where $b=\min\left\{n-a,(r-2)(k-1)\right\}$.

**Proof **

For simplicity of notation, set $\mathcal{H}=\mathcal{H}(n,k,a)$. We will first show that $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})\geq r(k-1)$. Then, we will prove that $\operatorname{alt}^{2}_{\sigma}(\mathcal{H})\geq a$, $\operatorname{alt}^{3}_{\sigma}(\mathcal{H})\geq a+1$, and $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})\geq a+b$ for $r\geq 4$, which clearly complete the proof. Consider an arbitrary bijection $\sigma:[n]\longrightarrow[n]$. Setting $X=(x_{1},\ldots,x_{n})$ with

[TABLE]

concludes $\operatorname{alt}(X)=r(k-1)$ and $|X^{i}|=k-1$ for each $i\in[r]$ implying $E(\mathcal{H}[\sigma(X^{i})])=\varnothing$ for each $i\in[r]$. Therefore, in view of the definition of $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})$, we have $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})\geq r(k-1)$. Since $\sigma$ is chosen arbitrarily, this implies $\operatorname{alt}^{r}(\mathcal{H})\geq r(k-1)$. Note that if $a\leq k-1$, then $\mathcal{H}=([n],{[n]\choose k})$. Now, for an arbitrary bijection $\sigma:[n]\longrightarrow[n]$, if $\operatorname{alt}(X)\geq r(k-1)+1$, then for at least one $j$, we must have $|X^{j}|\geq k$ which implies $E(\mathcal{H}[\sigma(X^{i})])\neq\varnothing$. This concludes $\operatorname{alt}^{r}(\mathcal{H})=\operatorname{alt}^{r}([n],{[n]\choose k})=r(k-1)$ completing the proof in this case. Henceforth, we suppose that $a\geq k$. Set $A=\{n-a+1,\ldots,n\}$. Let $\sigma:[n]\longrightarrow[n]$ be an arbitrary bijection. Also, let $\sigma^{-1}(A)=\{j_{1},\ldots,j_{a}\}$, where $1\leq j_{1}<\cdots<j_{a}\leq n$. The following three different cases will be distinguished.

• •

Case $\mathbf{r=2}:$ Define $X=(x_{1},\ldots,x_{n})$ such that

[TABLE]

Accordingly, we have $\operatorname{alt}(X)=|A|=a$ and $\sigma(X^{i})\subseteq A$ for each $i\in[r]$. This concludes $E(\mathcal{H}[\sigma(X^{i})])=\varnothing$ for each $i\in[r]$ implying that $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})\geq a$. Therefore, since $\sigma$ is arbitrary, we have $\operatorname{alt}^{r}(\mathcal{H})\geq a$. Note that we already proved that $\operatorname{alt}^{r}(\mathcal{H})\geq r(k-1)$ which completes the proof for $r=2$.

• •

Case $\mathbf{r=3:}$ Consider a fixed $i_{0}\in[n]\setminus A$. Set $X=(x_{1},\ldots,x_{n})$ such that

[TABLE]

Note $\operatorname{alt}(X)=a+1$ and moreover, $\sigma(X^{1}),\sigma(X^{2})\subseteq A$ and $|\sigma(X^{3})|=1$ which clearly implies $E(\mathcal{H}[\sigma(X^{i})])=\varnothing$ for each $i\in[r]$. Therefore, in view of the definition of $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})$, we have $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})\geq\operatorname{alt}(X)=a+1$. Since $\sigma$ is chosen arbitrarily, we have $\operatorname{alt}^{r}(\mathcal{H})\geq a+1$. Note that we have already proved that $\operatorname{alt}^{r}(\mathcal{H})\geq r(k-1)$, which completes the proof for $r=3$.

• •

Case $\mathbf{r\geq 4:}$ Consider a fixed $B=\{1,\ldots,b\}\subseteq[n]\setminus A$ such that $b=\min\left\{n-a,(r-2)(k-1)\right\}$. Also, let $\sigma^{-1}(B)=\{l_{1},\ldots,l_{b}\}$, where $1\leq l_{1}<\cdots<l_{b}\leq n$. Define $X=(x_{1},\ldots,x_{n})$ such that

[TABLE]

Clearly, $\operatorname{alt}(X)=|A|+b$ and $\sigma(X^{1}),\sigma(X^{2})\subseteq A$ and $|\sigma(X^{i})|\leq k-1$ for each $i\geq 3$ which conclude $E(\mathcal{H}[\sigma(X^{i})])=\varnothing$ for each $i\in[r]$. Therefore, in view of the definition of $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})$, we have $\operatorname{alt}^{r}_{\sigma}(\mathcal{H})\geq a+b$. Since $\sigma$ is chosen arbitrarily, we have $\operatorname{alt}^{r}(\mathcal{H})\geq\operatorname{alt}(X)=a+b$. This result in combination with the fact that $\operatorname{alt}^{r}(\mathcal{H})\geq r(k-1)$ implies the proof when $r\geq 4$.

$\blacksquare$

Suppose that $r\geq 4$, $a\geq rk-1$, and $n\geq(r-1)(k-1)+a$. For simplicity of notation, set $\mathcal{H}=\mathcal{H}(n,k,a)$. On the one hand, Observation 8 for $\boldsymbol{s}=(1,\ldots,1)$ yields $\operatorname{ecd}^{r}(\mathcal{H})=n-a$ and $\operatorname{cd}^{r}(\mathcal{H})=n-(r-1)(k-1)-a$, but on the other hand, Lemma  10 implies $n-\operatorname{alt}^{r}(\mathcal{H})\leq n-a-(r-2)(k-1)$. Therefore, $\operatorname{ecd}^{r}(\mathcal{H})-\operatorname{cd}^{r}(\mathcal{H})=(r-1)(k-1)$ and $\operatorname{ecd}^{r}(\mathcal{H})-(n-\operatorname{alt}^{r}(\mathcal{H}))\geq{(r-2)(k-1)},$ which clearly implies the following observation.

Observation 11

For $r\geq 4$, $a\geq rk-1$, and $n\geq(r-1)(k-1)+a$, the values of {1\over r-1}\big{(}\operatorname{ecd}^{r}(\mathcal{H})-\operatorname{cd}^{r}(\mathcal{H})\big{)} and {1\over r-1}\big{(}\operatorname{ecd}^{r}(\mathcal{H})-(n-\operatorname{alt}^{r}(\mathcal{H}))\big{)} can be made as large as desired by making $k$ large enough.

For the hypergraph $\mathcal{H}=\mathcal{H}(n,k,a)$, the following proposition demonstrates that $\left\lceil{\operatorname{ecd}^{r}(\mathcal{H})\over r-1}\right\rceil$ provides an exact lower bound for the chromatic number of $\operatorname{KG}^{r}(n,k,a)$ in some cases, while, in view of the previous observation, neither $\left\lceil{\operatorname{cd}^{r}(\mathcal{H})\over r-1}\right\rceil$ nor $\left\lceil{n-\operatorname{alt}^{r}(\mathcal{H})\over r-1}\right\rceil$ do that. Note that the following proposition can be considered as a generalization of the Alon-Frankl-Lovász theorem as well.

Proposition 12

Let $n,r,k,$ and $a$ be positive integers, where $r,k\geq 2$, $n>a$, and $n\geq rk$. Then we have

[TABLE]

provided that either $a\leq 2(k-1)$ or $a\geq rk-1$. Moreover, for $2k-1\leq a\leq rk-2$, we have

[TABLE]

**Proof **

The cases $a<k$ and $a\geq rk-1$ have been already proved by Corollary 9 when $\boldsymbol{s}=(1,\ldots,1)$. Also, by Lemma 7 and Observation 8, we have

[TABLE]

provided that $k\leq a\leq rk-2$. Therefore, to complete the proof, we must show that

[TABLE]

provided that $k\leq a\leq 2(k-1)$. Since $\operatorname{KG}^{r}(n,k,a)=\operatorname{KG}^{r}(\mathcal{H}(n,k,a))$, in view of Inequality 3, we have

[TABLE]

where $I:[n]\longrightarrow[n]$ is the identity bijection. Hence, we have the proof concluded if we prove that $\operatorname{alt}^{r}_{I}(\mathcal{H}(n,k,a))\leq r(k-1)$. To compute $\operatorname{alt}^{r}_{I}(\mathcal{H}(n,k,a))$, we must find the maximum possible value of $\operatorname{alt}(X)$ over all $X\in(Z_{r}\cup\{0\})^{n}$ such that none of $X^{i}$’s contains an edge of $\mathcal{H}(n,k,a)$. It is clear that this maximum also happens among $X$’s with $\operatorname{alt}(X)=|X|$, that is,

[TABLE]

Note $\operatorname{alt}(X)=|X|$ implies that none of $X^{i}$’s contains consecutive elements of $[n]$ and therefore, $|X^{i}\cap\{n-a+1,\ldots,n\}|\leq\lceil{a\over 2}\rceil\leq k-1$ for each $i$. Using this observation along with the fact that if $\operatorname{alt}(X)=|X|\geq r(k-1)+1$, then there is at least one $i\in[r]$ for which $|X^{i}|\geq k$ simply concludes $\operatorname{alt}^{r}_{I}(\mathcal{H}(n,k,a))\leq r(k-1)$ as desired.

• $\blacksquare$*

We here pose the following conjecture asserting that the hypothesis $a\leq 2(k-1)$ or $a\geq rk-1$ is superfluous in the preceding proposition.

Conjecture 13

Let $n,r,k,$ and $a$ be positive integers, where $r,k\geq 2$, $n>a$, and $n\geq rk$. We have

[TABLE]

This conjecture is strongly supported by Proposition 12. To be more specific, note that the validity of this conjecture is already verified for $a\leq 2(k-1)$, $a\geq rk-1$, and, in particular, for $r=2$ by Proposition 12. Therefore, it is left open just for $2k-1\leq a\leq rk-2$ with $r\geq 3$. Even more, if $a\leq r(k-1)$ and $r$ is a power of $2$, then we can deduce Conjecture 13 from a result by Alon, Drewnowski, and Łuczak [3]. For $t\geq 2$, a subset $A$ of $[n]$ is called $t$ stable if $t\leq|x-y|\leq n-t$ for each $x\neq y\in A$. The induced subhypergraph of $\operatorname{KG}^{r}(n,k)$ by the set of all $t$-stable vertices is called the $t$-stable Kneser hypergraph $\operatorname{KG}^{r}(n,k)_{t-stab}$. Alon, Drewnowski, and Łuczak [3] proved that $\chi(\operatorname{KG}^{r}(n,k)_{r-stab})=\chi(\operatorname{KG}^{r}(n,k))$ provided that $r$ is a power of $2$. They also conjectured that this result is true for general $r$. One can simply see that the hypergraph $\operatorname{KG}^{r}(n,k,a)$ in the statement of Conjecture 13 is a super-hypergraph of $\operatorname{KG}^{r}(n,k)_{r-stab}$ provided that $a\leq r(k-1)$, which implies that Conjecture 13 is true provided that $a\leq r(k-1)$ and $r$ is a power of $2$ . In this point of view, we can consider this conjecture as a weak version of the Alon-Drewnowski-Łuczak conjecture.

For a hypergraph $\mathcal{H}$, though the lower bound of $\chi(\operatorname{KG}^{r}(\mathcal{H}))$ which is based on $\operatorname{ecd}^{r}(\mathcal{H})$ might be appreciably better than the lower bound based on $\operatorname{alt}^{r}(\mathcal{H})$, in general we cannot decide which one is better. There are several examples in which the lower bound of $\chi(\operatorname{KG}^{r}(\mathcal{H}))$ based on $\operatorname{alt}^{r}(\mathcal{H})$ is much better than the lower bound gained by $\operatorname{ecd}^{r}(\mathcal{H})$. As an instance, for positive integers $n$, $r$, and $k$, let $K=K_{n,...,n}$ be a graph with vertex set $[rn]$ and edge set $E(K)=\left\{e\in{[rn]\choose 2}:e\nsubseteq\{1+(t-1)n,...,tn\}\quad\forall t\in[r]\right\}$. By the definition of $K$, it is observed that $V(K)$ is partitioned into $r$ pairwise disjoint sets of size $n$ such that none of them contains any edge of $K$. So, $\operatorname{ecd}^{r}(K)=0$, while it can be verified that

[TABLE]

where $I:[rn]\longrightarrow[rn]$ is the identity bijection.

Proposition 14

Let $r$ be a fixed integer where $r\geq 2$. For a given hypergraph $\mathcal{F}$, it is notable that though $\left\lceil{\operatorname{ecd}^{r}(\mathcal{F})\over r-1}\right\rceil$ provides a lower bound for the chromatic number of $\operatorname{KG}^{r}(\mathcal{F})$, computing $\operatorname{ecd}^{r}(\mathcal{F})$ is an $NP$-hard problem.

The proof of this observation is almost the same as that of Proposition 6 in [17]. However, for the sake of completeness, we sketch the proof in the following.

Let $r$ be a fixed integer, where $r\geq 2$. Also, let $G$ be a given graph for which we want to compute the independence number $\alpha(G)$. Define $\bar{G}$ to be a graph which is constructed by the union of $r$ vertex-disjoint copies of $G$ such that any two vertices from different copies are adjacent. It is not difficult to see that

[TABLE]

This observation implies that computing the equitable colorability defect of graphs is at least as difficult as computing the independence number of graphs. Since computing the independence number of graphs is known as an NP-hard problem, we have the proof completed.

Remark 15

After this paper became available online, using different methods, Aslam et al. [5] proved Conjecture 13 in some cases.

Acknowledgments

We would like to thank professor Frédéric Meunier for his comments that helped to improve the presentation of the paper. Also, we thank the two anonymous reviewers for their careful reading of our manuscript and their constructive comments.