Stability results on vertex Tur\'an problems in Kneser graphs

TL;DR

This paper establishes stability results for vertex Turán problems in Kneser graphs, determining second largest independent sets avoiding specific subgraphs like even cycles and multipartite graphs, generalizing classical theorems.

Contribution

It extends known results by identifying second largest $F$-free vertex sets in Kneser graphs for certain forbidden subgraphs, based on their chromatic number.

Findings

Determined second largest $F$-free vertex sets for even cycles and multipartite graphs.

Generalized classical theorems like Erdős-Ko-Rado and Hilton-Milner.

Provided stability results for vertex Turán problems in Kneser graphs.

Abstract

The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free, was recently determined by Alishahi and Taherkhani, whenever $n$ is large enough compared to $k$ and $F$. In this paper, we determine the second largest size of a vertex set $W \subseteq V$ such that $K(n,k)[W]$ is $F$-free, in the case when $F$ is an even cycle or a complete multi-partite graph. In the latter case, we actually give a more general theorem depending on the chromatic number of $F$. These results generalize the celebrated Erd\H os-Ko-Rado theorem and Hilton-Milner theorem.