Extremal $G$-free induced subgraphs of Kneser graphs

Abstract

The Kneser graph ${\rm KG}_{n,k}$ is a graph whose vertex set is the family of all $k$-subsets of $[n]$ and two vertices are adjacent if their corresponding subsets are disjoint. The classical Erd\H{o}s-Ko-Rado theorem determines the cardinality and structure of a maximum induced $K_2$-free subgraph in ${\rm KG}_{n,k}$. As a generalization of the Erd\H{o}s-Ko-Rado theorem, Erd\H{o}s proposed a conjecture about the maximum order of an induced $K_{s+1}$-free subgraph of ${\rm KG}_{n,k}$. As the best known result concerning this conjecture, Frankl [Journal of Combinatorial Theory, Series A, 2013], when $n\geq(2s+1)k-s$, gave an affirmative answer to this conjecture and also determined the structure of such a subgraph. In this paper, generalizing the Erd\H{o}s-Ko-Rado theorem and the Erd{\H o}s matching conjecture, we consider the problem of determining the structure of a maximum family $\mathcal{A}$ for which ${\rm KG}_{n,k}[\mathcal{A}]$ has no subgraph isomorphic to a given graph $G$. In this regard, we determine the size and the structure of such a family provided that $n$ is sufficiently large with respect to $G$ and $k$. Furthermore, for the case $G=K_{1,t}$, we present a Hilton-Milner type theorem regarding above-mentioned problem, which specializes to an improvement of a result by Gerbner et al. [SIAM Journal on Discrete Mathematics, 2012].