Extremal G-free induced subgraphs of Kneser graphs

The Kneser graph $\KG_{n,k}$ 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 $\KG_{n,k}$. 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 $\KG_{n,k}$. As the best known result concerning this conjecture, Frankl~[{\it 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 $\A$ for which $\KG_{n,k}[\A]$ 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 and et al.~[{\it SIAM Journal on Discrete Mathematics, 2012.}]