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arXiv · 1804.03988

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

Dániel Gerbner·Abhishek Methuku·Dániel Nagy·Balázs Patkós·Máté Vizer
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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.

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Dániel Gerbner, Abhishek Methuku, Dániel Nagy, Balázs Patkós, Máté Vizer. 2018-04-11. Stability results on vertex Tur\'an problems in Kneser graphs. https://arxiv.org/abs/1804.03988

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