  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].

Combinatorics