On "stability" in the Erd\H{o}s-Ko-Rado theorem

Pat Devlin; Jeff Kahn

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arXiv · 1502.05692

On "stability" in the Erd\H{o}s-Ko-Rado theorem

Pat Devlin·Jeff Kahn

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Abstract

Denote by $K_p(n,k)$ the random subgraph of the usual Kneser graph $K(n,k)$ in which edges appear independently, each with probability $p$. Answering a question of Bollob\'as, Narayanan, and Raigorodskii,we show that there is a fixed $p<1$ such that a.s. (i.e., with probability tending to 1 as $k \to \infty$) the maximum independent sets of $K_p(2k+1, k)$ are precisely the sets $\{A\in V(K(2k+1,k)): x\in A\}$ ($x\in [2k+1]$). We also complete the determination of the order of magnitude of the "threshold" for the above property for general $k$ and $n\geq 2k+2$. This is new for $k\sim n/2$, while for smaller $k $ it is a recent result of Das and Tran.

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Pat Devlin, Jeff Kahn. 2015-02-19. On "stability" in the Erd\H{o}s-Ko-Rado theorem. https://arxiv.org/abs/1502.05692

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