More non-bipartite forcing pairs

TL;DR

This paper investigates pairs of graphs called forcing pairs, focusing on non-bipartite cases, and establishes that a specific number of doublings always suffices to ensure quasirandomness.

Contribution

It proves that for any t>3, the minimum number of doublings needed for forcing pairs is always (t+2)/2, extending previous results.

Findings

(K_t,F) is forcing where F is obtained by iterative doubling of K_t

Two doublings suffice for t=3 to establish forcing pairs

For all t>3, (t+2)/2 doublings are enough

Abstract

We study pairs of graphs (H_1,H_2) such that every graph with the densities of H_1 and H_2 close to the densities of H_1 and H_2 in a random graph is quasirandom; such pairs (H_1,H_2) are called forcing. Non-bipartite forcing pairs were first discovered by Conlon, Han, Person and Schacht [Weak quasi-randomness for uniform hypergraphs, Random Structures Algorithms 40 (2012), 1-38]: they showed that (K_t,F) is forcing where F is the graph that arises from K_t by iteratively doubling its vertices and edges in a prescribed way t times. Reiher and Schacht [Forcing quasirandomness with triangles, Forum of Mathematics, Sigma 7, 2019] strengthened this result for t=3 by proving that two doublings suffice and asked for the minimum number of doublings needed for t>3. We show that (t+2)/2 doublings always suffice.

Equations

$$t(F_1,G)=(1\pm \delta )p^{e(F_1)}\text{and}t(F_2,G)=(1\pm \delta )p^{e(F_2)}$$

$$t(F,W)={\int }_{[0,1{]}^{V(F)}}\prod _{uv\in E(F)}W(x_u,x_v)\prod _{u\in V(F)}d{x}_u.$$

$$t(K_4,W)=p^6\text{and}t({\mathcal{T}}_3(K_4),W)=p^{48}.$$

$$W(x_1,x_2)W(x_1,x_3)W(x_2,x_3){\int }_{[0,1]}W(x_1,y)W(x_2,y)W(x_3,y)dy=p^6$$

$$W(x_1,x_2)W(x_1,x_3)W(x_2,x_3){\int }_{[0,1]}W(x_1,y)W(x_2,y)W(x_3,y)dy=p^6.$$

$$W(x_1,x_2)W(x_1,x_3)W(x_2,x_3){\int }_{[0,1]}W(x_1,y)W(x_2,y)W(x_3,y)dy=p^6$$

$$f(x)=\underset{y_1,y_2\in [0,1]}{\sup }(W(x,y_1)-W(x,y_2)).$$

$$W(x_1,x_2)$$

$$W(x_2,x_4)$$

$$W(x_1,y)\ge a-\eta \text{and}W(x_4,y)\le b+\eta \text{for almost all }y\in [0,1].$$

$$W(x_1,x_2)W(x_1,x_3)W(x_2,x_3){\int }_{[0,1]}W(x_1,y)W(x_2,y)W(x_3,y)dy$$

$$=p^6=W(x_2,x_4)W(x_3,x_4)W(x_2,x_3){\int }_{[0,1]}W(x_4,y)W(x_2,y)W(x_3,y)dy$$

$$(b-\eta {)}^2(a-\eta )W(x_2,x_3){\int }_{[0,1]}W(x_2,y)W(x_3,y)dy$$

$$(a+\eta {)}^2(b+\eta )W(x_2,x_3){\int }_{[0,1]}W(x_2,y)W(x_3,y)dy.$$

$$(b-\eta {)}^2(a-\eta )\le (a+\eta {)}^2(b+\eta )$$

$$t(K_t,W)=p^{e(K_t)}\text{and}t({\mathcal{T}}_k(K_t),W)=p^{e({\mathcal{T}}_k(K_t))}.$$

$$p^{e(K_t)}=$$

$$\cdot {\int }_{[0,1{]}^{t-k}}\left(\prod _{1\le i<j\le t-k}W(y_i,y_j)\right)\left(\prod _{i\in [k],j\in [t-k]}W(x_i,y_j)\right)\prod _{i\in [t-k]}d{y}_i.$$

$$p^{e(K_t)}$$

$$\cdot {\int }_{[0,1{]}^{t-k}}\left(\prod _{1\le i<j\le t-k}W(y_i,y_j)\right)\left(\prod _{i\in [k],j\in [t-k]}W(x_i,y_j)\right)\prod _{i\in [t-k]}d{y}_i.$$

$$f(x)=\underset{y_1,y_2\in [0,1]}{\sup }(W(x,y_1)-W(x,y_2)).$$

$$W(x_1,x_i)\ge b-\eta \text{and}W(x_{k+1},x_i)\le a+\eta \text{for all }i=2,\dots ,k,$$

$$W(x_1,y),W(x_{k+1},y)\in [a-\eta ,b+\eta ]\text{ for almost all }y\in [0,1],$$

$$Q=$$

$$\cdot {\int }_{[0,1{]}^{t-k}}\left(\prod _{1\le i<j\le t-k}W(y_i,y_j)\right)\left(\prod _{2\le i\le k,j\in [t-k]}W(x_i,y_j)\right)\prod _{i\in [t-k]}d{y}_i.$$

$$\left(\prod _{1\le i<j\le k}W(x_i,x_j)\right)$$

$$\cdot {\int }_{[0,1{]}^{t-k}}\left(\prod _{1\le i<j\le t-k}W(y_i,y_j)\right)\left(\prod _{1\le i\le k,j\in [t-k]}W(x_i,y_j)\right)\prod _{i\in [t-k]}d{y}_i=p^{e(K_t)}$$

$$=$$

$$\cdot {\int }_{[0,1{]}^{t-k}}\left(\prod _{1\le i<j\le t-k}W(y_i,y_j)\right)\left(\prod _{2\le i\le k+1,j\in [t-k]}W(x_i,y_j)\right)\prod _{i\in [t-k]}d{y}_i.$$

$$(b-\eta {)}^{k-1}(a-\eta {)}^{t-k}Q$$

$$(a+\eta {)}^{k-1}(b+\eta {)}^{t-k}Q.$$

$$(b-\eta {)}^{k-1}(a-\eta {)}^{t-k}\le (a+\eta {)}^{k-1}(b+\eta {)}^{t-k}.$$

$$W(x_1,y)\le a+2t\eta \text{and}W(x_{k+1},y)\ge b-2t\eta \text{for almost all }y\in [0,1].$$

$$W(x_1,y)\in [a\pm 2t\eta ]\text{and}W(x_{k+1},y)\in [b\pm 2t\eta ]\text{for almost all }y\in [0,1].$$

Full text

1 Introduction and results

The systematic study of quasirandom graphs has been initiated by Thomason [14, 15] and Chung, Graham and Wilson [1] in the 1980’s. Since then, many properties of quasirandom graphs were described. We refer to the surveys [6, 7].

A key property of a quasirandom graph is an almost uniform edge distribution. A sequence $(G_{n})_{n\in\mathbb{N}}$ of graphs is $p$-quasirandom if $e_{G_{n}}(U)=p\binom{|U|}{2}+o(|G_{n}|^{2})$ for all subsets $U\subseteq V(G_{n})$, where $|G_{n}|$ is the number of vertices of $G_{n}$ and $e_{G_{n}}(U)$ is the number of edges of $G_{n}$ with both end vertices in $U$. A particular graph $G$ with $n$ vertices is $(\varepsilon,p)$-quasirandom if $\left|e_{G}(U)-p\binom{|U|}{2}\right|\leq\varepsilon n^{2}$ for all subsets $U\subseteq V(G)$.

One of many equivalent characterizations of $p$-quasirandom sequences $(G_{n})_{n\in\mathbb{N}}$ of graphs is the following: $(G_{n})_{n\in\mathbb{N}}$ is quasirandom if and only if $G_{n}$ has edge density $p$ and contains $(p^{4}+o(1))|G_{n}|^{4}$ labelled (non-induced) copies of $C_{4}$. Equivalently, $G_{n}$ contains asymptotically the expected number of copies of $K_{2}$ and $C_{4}$ as the Erdős-Rényi random graph $G(n,p)$. This leads to the definition of a forcing pair of graphs given below. To give the definition, we need to introduce the following notation. If $F$ and $G$ are two graphs, then $t(F,G)$ is the number of graph homomorphisms from $F$ to $G$, i.e. all maps $f:V(F)\rightarrow V(G)$ with $f(e)\in E(G)$ for all $e\in E(F)$. In addition, we write $e(F)$ for the number of edges of $F$.

Definition 1.1** (Forcing pairs).**

A pair $(F_{1},F_{2})$ is called forcing if for every $p\in(0,1]$ and $\varepsilon>0$, there exists a $\delta>0$ such that the following holds. Every graph $G$ with

[TABLE]

is $(\varepsilon,p)$-quasirandom.

In particular, the pair $(K_{2},C_{4})$ is forcing. There are two exciting conjectures related to forcing pairs: Sidorenko’s conjecture made independently by Sidorenko [11] and by Erdős and Simonovits [12], and the so-called forcing conjecture made by Skokan and Thoma [13]. While Sidorenko’s conjecture asks whether the lower bound on $t(F_{2},G)$ is always at least $p^{e(F_{2})}$ where $p=t(K_{2},G)$, the forcing conjecture states that any pair $(K_{2},F)$, where $F$ is a bipartite graph containing a cycle, is forcing. Due to their relation to Szemerédi’s regularity lemma, these conjectures expedited tremendous amount of research in extremal combinatorics. Thus, additional forcing pairs were studied in [1, 2, 3, 4, 5, 9, 13]. The first non-bipartite forcing pairs were found by Conlon et al. in [3]. So far, all non-bipartite forcing pairs are obtained by the construction described below.

Let $F$ be a $t$-partite graph and $V(F)=V_{1}(F)\dot{\cup}\ldots\dot{\cup}V_{t}(F)$ a fixed $t$-coloring of $F$. The doubling $\mathcal{T}(F)$ on $V_{1}(F)$ is the graph obtained by taking two identical disjoint copies $F_{1}$ and $F_{2}$ of $F$ and identifying the corresponding vertices in $V_{1}(F_{1})$ and $V_{1}(F_{2})$. In this way, we obtain a $t$-coloring of $\mathcal{T}(F)$ given by the sets $V_{1}(F_{1})=V_{1}(F_{2})$, $V_{2}(F_{1})\dot{\cup}V_{2}(F_{2})$, …, $V_{t}(F_{1})\dot{\cup}V_{t}(F_{2})$. For $k\leq t$, the $k$-fold doubling $\mathcal{T}_{k}(F)$ is defined as the doubling $\mathcal{T}(F)$ on $V_{1}(F)$ for $k=1$ and the doubling $\mathcal{T}(\mathcal{T}_{k-1}(F))$ on $V_{k}(\mathcal{T}_{k-1}(F))$ for $k\geq 2$. The order of the doublings has no influence on $\mathcal{T}_{k}(F)$, i.e. we could permute $V_{1}(F),\dots,V_{k}(F)$ arbitrarily. Observe that $\mathcal{T}_{2}(K_{2})=C_{4}$.

The pair $(K_{2},C_{4})=(K_{2},\mathcal{T}_{2}(K_{2}))$ is forcing. The result from [3] states that the pair $(K_{t},\mathcal{T}_{t}(K_{t}))$ is also forcing for any $t\geq 3$. Hàn et al. [4] generalized this result for any $t$-colorable graph $F$ in a similar way. Reiher and Schacht [9] improved the result from [3] for $t=3$ by showing that the pair $(K_{3},\mathcal{T}_{2}(K_{3}))$ is forcing. We generalize this result for $t>3$; we note that the same was independently proven by Reiher and Schacht [10].

Theorem 1.2**.**

The pair $(K_{t},\mathcal{T}_{\lceil(t+1)/2\rceil}(K_{t}))$ is forcing for any $t\geq 2$.

We will present the proof in the language of graph limits, which we now introduce, since this makes the arguments particularly short and transparent. Let $W$ be a kernel, i.e. a bounded symmetric Lebesgue measurable function from $[0,1]^{2}$. We write $W\equiv p$ if $W$ is equal to $p$ almost everywhere; a kernel with $W\equiv p$ is called $p$-quasirandom. The homomorphism density extends in a natural way for a graph $F$ and a kernel $W$:

[TABLE]

A graphon is a kernel $W$ with values in $[0,1]$. A pair of graphs $(F_{1},F_{2})$ is called forcing if for every real $p\in(0,1]$, every graphon $W$ with $t(F_{1},W)=p^{e(F_{1})}$ and $t(F_{2},W)=p^{e(F_{2})}$ is $p$-quasirandom. This definition, see [8, Chapter 16], coincides with the definition of a forcing pair given earlier.

2 Proof of Theorem 1.2 for $t=4$

In this section, we give a proof of Theorem 1.2 for $K_{4}$. We need the following lemma.

Lemma 2.1** (Lemma 10 from [3]).**

Let $W$ be a graphon and $p\in(0,1]$ such that

[TABLE]

Then

[TABLE]

for almost all $(x_{1},x_{2},x_{3})\in[0,1]^{3}$.∎

The proof of this lemma is given in [3] in the language of quasirandom (hyper)graphs, and we sketch the line of arguments here for completeness. It can be proved by repeatedly applying Cauchy-Schwarz inequality starting to $t(K_{4},W)$ three times. This series of applications of Cauchy-Schwarz inequality yields that $t(\mathcal{T}_{3}(K_{4}),W)\geq p^{48}$. Since it holds $t(\mathcal{T}_{3}(K_{4}),W)=p^{48}$ by the assumption of the lemma, it follows that for almost all values of $(x_{1},x_{2},x_{3})\in[0,1]^{3}$ one has

[TABLE]

The next lemma together with Lemma 2.1 readily implies Theorem 1.2 for $k=4$.

Lemma 2.2**.**

Let $W$ be a graphon and $p\in(0,1]$. If it holds that

[TABLE]

for almost all $(x_{1},x_{2},x_{3})\in[0,1]^{3}$, then $W$ is $p$-quasirandom.

Before presenting the proof, we recall the definition of the essential supremum $\operatorname{ess}\sup(f)$ of a (Lebesgue) measurable function $f\colon\mathbb{R}^{n}\to\mathbb{R}$. It is the infimum over all $y\in\mathbb{R}$ with $f(x)\leq y$ for almost all $x\in\mathbb{R}^{n}$, i.e. $\operatorname{ess}\sup(f):=\inf\left\{y\colon\lambda(\{f\geq y\})=0\right\}$, where $\lambda$ is the Lebesgue measure. The essential infimum of a function is defined analogously.

Proof of Lemma 2.2.

Let $f:[0,1]\rightarrow[0,1]$ be defined as

[TABLE]

Observe that $f$ is a measurable function and set $c:=\operatorname{ess}\sup f\in[0,1]$. If $c=0$, then $W$ is $p$-quasirandom. Thus, we assume that $c>0$.

The definition of $f$ and $c$ implies that there exist reals $a,b\in\mathbb{R}$ with $c=b-a$ satisfying the following. For any $\eta>0$, there exist $x_{1},x_{2},x_{3},x_{4}\in[0,1]$ such that

[TABLE]

and (1) holds for $(x_{1},x_{2},x_{3})$ and $(x_{2},x_{3},x_{4})$. In addition, we can assume that $a>0$, $W(x_{2},x_{3})>0$, $\operatorname{ess}\sup_{y\in[0,1]}W(x_{2},y)W(x_{3},y)>0$ (because $p>0$), and that

[TABLE]

We get from (1) that

[TABLE]

Using (2) and (3) we can lower bound the left hand side by

[TABLE]

and similarly we can upper bound the right hand side by

[TABLE]

As $W(x_{2},x_{3})\int_{[0,1]}W(x_{2},y)W(x_{3},y)\operatorname{d}y$ is non-zero, we obtain that

[TABLE]

and then deduce that $b\leq a+10\eta(b/a)$. Together with the assumption that $b>a$ this implies that $W(x,y)\in[p\pm 20\eta(b/a)]$ for almost all $(x,y)\in[0,1]^{2}$. Since this holds for every $\eta>0$, the lemma follows. ∎

3 Proof of Theorem 1.2—general case

The proof of the general case is based on the same idea as used in the previous section and follows from the next two lemmas. The first lemma can be proven by repeated applications of the Cauchy-Schwarz inequality similarly to the proof of Lemma 2.1.

Lemma 3.1**.**

Let $W$ be a graphon, $p\in(0,1]$, $t\geq 3$, and $k:=\lceil(t+1)/2\rceil$ such that

[TABLE]

Then

[TABLE]

for almost all $(x_{1},\dots,x_{k})\in[0,1]^{k}$.

Lemma 3.2**.**

Let $W$ be a graphon, $p\in(0,1]$, $t\geq 3$, and $k:=\lceil(t+1)/2\rceil$. If it holds that

[TABLE]

for almost all $(x_{1},\dots,x_{k})\in[0,1]^{k}$, then $W$ is $p$-quasirandom.

Proof.

Let $f:[0,1]\rightarrow[0,1]$ be defined as

[TABLE]

Again, $f$ is a measurable function and we set $c:=\operatorname{ess}\sup f\in[0,1]$. If $c=0$, then $W$ is $p$-quasirandom. So, we assume that $c>0$ and consider positive reals $a,b\in\mathbb{R}$ with $c=b-a$ such that the following holds. For any $\eta>0$, there exist $x_{1},\dots,x_{k+1}\in[0,1]$ such that

[TABLE]

for $(x_{1},\dots,x_{k})$ and $(x_{2},\dots,x_{k+1})$ equation (4) holds,

[TABLE]

and $Q$ is non-zero, where

[TABLE]

We get from (4) that

[TABLE]

Using (5) and (6) we can lower bound the left hand side by

[TABLE]

and similarly we can upper bound the right hand side by

[TABLE]

As $Q$ is non-zero this gives

[TABLE]

If $t$ is even, we have $k-1>t-k$ and we can finish the proof similar to the case $t=4$. If $t$ is odd, i.e. $k-1=t-k$, a more refined argument is needed.

Since $W$ is a graphon, the difference of (7) and (8) is at most $2t\eta$. In particular, the estimates used to derive (7) cannot be too wasteful and it follows that

[TABLE]

Together with (6) we get

[TABLE]

It follows that $b-a\leq 2(t+1)\eta$ and consequently $W(x,y)\in[p\pm 4(t+1)\eta]$ for almost all $(x,y)\in[0,1]^{2}$. ∎