Clique number of Xor products of Kneser graphs

TL;DR

This paper studies the clique number of the XOR product of Kneser graphs, providing bounds, exact values for specific cases, and asymptotic behavior, linking graph theory with extremal set theory.

Contribution

It offers new bounds and exact solutions for the clique number of XOR products of Kneser graphs, advancing understanding in graph and extremal set theory.

Findings

Derived bounds for f(k,N)

Exact value of f(2,N) for large N

Asymptotic equivalence of f(k,k^2) to k^2

Abstract

In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in "A general 2-part Erd\H{o}s-Ko-Rado theorem" by Gyula O.H. Katona, who proposed our problem as well. In the graph theoretic form we examine the clique number of the Xor product of two isomorphic $KG(N,k)$ Kneser graphs. Denote this number with $f(k,N)$. We give lower and upper bounds on $f(k,N)$, and we solve the problem up to a constant deviation depending only on $k$, and find the exact value for $f(2,N)$ if $N$ is large enough. We also compute that $f(k,k^2)$ is asymptotically equivalent to $k^2$.