# 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.

## Key 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.