On Tur\'{a}n problems for Cartesian products of graphs

TL;DR

This paper investigates Turán-type problems for Cartesian products of graphs, establishing asymptotic bounds for the minimum size of certain quadruple families that cover specific subsets, advancing understanding in extremal combinatorics.

Contribution

It provides the first asymptotic solution to a Turán problem involving Cartesian products and quadruples, extending classical extremal graph theory results.

Findings

Asymptotic minimum size of quadruple family is (1/16 + o(1)) n^2 m^2.

Solved a more general two-partite Turán problem for quadruples.

Established bounds as n,m tend to infinity.

Abstract

Let $A,B$ be disjoint sets of sizes $n$ and $m$. Let ${\mathcal Q}$ be a family of quadruples, having $2$ elements from $A$ and $2$ from $B$, such that any subset $S \subseteq A \cup B$ with $|S|=7$, $|S \cap A| \geq 2$ and $|S \cap B| \geq 2$ contains one of the quadruples. We prove that the smallest size of ${\mathcal Q}$ is $(1/16 + O(1/n) + O(1/m)) n^2 m^2$ as $n,m\to\infty$. We also solve asymptotically a more general two-partite Tur\'{a}n problem for quadruples.

Equations

$$T(n,k,2)=mn-\frac{m(m+1)}{2}(k-1)\mathrm{if}m\le \frac{n}{k-1}\le m+1.$$

Full text

An $r$-graph is a pair $H=(V(H),E(H))$ where $V(H)$ is a finite set of vertices, and the edge set $E(H)$ is a collection of $r$-subsets of $V(H)$. A subset of vertices is called independent if it contains no edges of $H$. The independence number $\alpha(H)$ is the largest size of an independent subset.

The classical Turán number $T(n,k,r)$ is the minimum number of edges in an $n$-vertex $r$-graph $H$ with $\alpha(H)<k$. Consequently, $\binom{n}{r}-T(n,k,r)$ is the largest number of edges in an $n$-vertex $r$-graph that does not contain a complete subgraph on $k$ vertices. The ratio $T(n,k,r)/\binom{n}{r}$ is non-decreasing when $n$ increases, so the limit $t(k,r)=\lim_{n\to\infty}T(n,k,r)/\binom{n}{r}$ exists.

The exact values of $T(n,k,2)$ were found by Mantel [5] for $k=3$, and by Turán [9] for all $k$:

[TABLE]

In particular, $t(k,2)=1/(k-1)$. Not a single value $t(k,r)$ is known with $k>r>2$.

Giraud [4] discovered an elegant construction for $r=4$, $k=5$ which yields $t(5,4)\leq 5/16$. This construction was generalized by de Caen, Kreher and Wiseman [1] in the following way. Consider two disjoint sets $A=\{a_{1},a_{2},\ldots,a_{n}\}$, $B=\{b_{1},b_{2},\ldots,b_{m}\}$, and a $0,1$-matrix $X=[x_{ij}]$ of size $n\times m$. Let $E_{40}$ be the set of all quadruples within $A$, $E_{04}$ be the set of all quadruples within $B$, and $E_{22}$ be the set of quadruples $\{a_{i},a_{j},b_{p},b_{q}\}$ such that $x_{ip}+x_{iq}+x_{jp}+x_{jq}$ is even. It is easy to see that in the $4$-graph $H=(A\cup B,\>E_{40}\cup E_{04}\cup E_{22})$ any subset of $5$ vertices contains at least one edge. If $n$ and $m$ are approximately equal, and the entries of $X$ are selected randomly and independently, with equal probability of being [math] or $1$, the expected number of edges in $H$ is $\frac{5}{16}\binom{n+m}{4}+O((n+m)^{3})$. A more specific choice of $X$ related to Hadamard matrices provides the best known upper bounds for $T(n,5,4)$ (see [7]).

If $n,m\to\infty$, then \big{(}\frac{1}{2}+o(1)\big{)}\binom{n}{2}\binom{m}{2} is the minimal size of a system of quadruples with $2$ elements from $A$ and from $B$ such that every quintuple with $3$ elements from $A$ and $2$ from $B$, or vice versa, contains at least one of the quadruples.

In [2], de Caen, Kreher and Wiseman defined a broader set of Turán type problems which we describe here for quadruples setting only. Let $Q(n,m,a,b)$ denote the smallest size of a system of quadruples with $2$ elements from $A$ and from $B$ such that every $(a+b)$-set with $a$ elements from $A$ and $b$ from $B$ contains at least one of the quadruples. Obviously, $Q(n,m,a,b)\leq T(n,a,2)\cdot T(m,b,2)$. It was proved in [2] that $Q(n,n,3,3)=q\binom{n}{2}^{2}+o(n^{4})$ where $1/4\geq q\geq(3-\sqrt{5})/4\approx 0.1910$.

In this note, we consider the following problem. Let ${\mathcal{P}}$ be a set of pairs $(a,b)$ with $a,b\geq 2$. Let $A$ and $B$ be disjoint sets with $|A|=n$ and $|B|=m$. Denote by $Q(n,m,{\mathcal{P}})$ the minimum size of a system of quadruples with $2$ elements from $A$ and from $B$ such that for every $(a,b)\in{\mathcal{P}}$, every $(a+b)$-set with $a$ elements from $A$ and $b$ from $B$ contains at least one of the quadruples. The cases ${\mathcal{P}}=\{(2,3),(3,2)\}$ and ${\mathcal{P}}=\{(3,3)\}$ were studied in [1] and [2], respectively. We will focus on the cases ${\mathcal{P}}_{k2}=\{(2,k+1),(k+1,2)\}$ and ${\mathcal{P}}_{k3}=\{(2,k+1),(3,k),(k,3),(k+1,2)\}$. To solve them, we need to generalize the above-mentioned construction with $2\times 2$ submatrices.

Definition. A $2\times 2$ matrix $[x_{ij}]$ over an additive abelian group is called fair if $x_{11}+x_{22}=x_{12}+x_{21}$.

Let $G_{k}$ be a graph whose vertices are functions $f:\mathbb{Z}_{k}\to\mathbb{Z}_{k}\>$. A pair of vertices $\{f,g\}$ forms an edge in $G_{k}$ if $f-g$ is a bijection. Lemma 2 restates the fact that $G_{k}$ has no triangles when $k$ is even. For odd $k$, the problem of counting triangles in $G_{k}$ has been solved asymptotically in [3]. Let $p(k)$ be the smallest prime factor of $k$. The $p(k)$ functions $f_{0},f_{1},\ldots,f_{p(k)-1}$, where $f_{i}(j)=i\cdot j\pmod{k}$, form a complete subgraph in $G_{k}$. It is very tempting to conjecture that $p(k)$ is indeed the size of the largest clique in $G_{k}$. We know that this is true for even $k$ and for prime $k$. Computer search confirms that this is also true for $k=9$.

To prove the upper bound, consider an $n\times m$ matrix $X=[x_{ij}]$ over $\mathbb{Z}_{k}$. Let $A=\{a_{1},a_{2},\ldots,a_{n}\}$, $B=\{b_{1},b_{2},\ldots,b_{m}\}$, and let ${\mathcal{Q}}$ consist of quadruples $\{a_{i},a_{j},b_{p},b_{q}\}$ such that rows $i,j$ and columns $p,q$ in $X$ produce a fair $2\times 2$ submatrix. By Lemma 1, $Q(n,m,{\mathcal{P}}_{k2})\leq|{\mathcal{Q}}|$, and if $k$ is even, by Lemma 2, $Q(n,m,{\mathcal{P}}_{k3})\leq|{\mathcal{Q}}|$. If entries of $X$ are selected randomly, independently and uniformly over $\mathbb{Z}_{k}$, the expected value of $|{\mathcal{Q}}|$ is $\frac{1}{k}\binom{n}{2}\binom{m}{2}$ which provides the required upper bound. ∎

The result mentioned in the abstract follows from Theorem 3 when $k=4$ and ${\mathcal{P}}={\mathcal{P}}_{43}$.

Turán problems, where the extremal configurations depend on random maps defined on the set of pairs of vertices, have been studied in a recent series of articles by Rödl and his coauthors (see the concluding remarks in [6]). Another “partite” version of Turán problem and its connection to the classical problem has been studied by Talbot [8].

Acknowledgments

The author would like to thank two anonymous referees for their careful reading and valuable suggestions.