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arXiv · 2105.10971

Independent sets in subgraphs of a shift graph

Andrii Arman·Vojtěch Rödl·Marcelo Tadeu Sales
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Abstract

Erdős, Hajnal and Szemerédi proved that any subset $G$ of vertices of a shift graph $\text{Sh}_{n}^{k}$ has the property that the independence number of the subgraph induced by $G$ satisfies $α(\text{Sh}_{n}^{k}[G])\geq \left(\frac{1}{2}-\varepsilon\right)|G|$, where $\varepsilon\to 0$ as $k\to \infty$. In this note we prove that for $k=2$ and $n \to \infty$ there are graphs $G\subseteq \binom{[n]}{2}$ with $α(\text{Sh}_{n}^{2}[G])\leq \left(\frac{1}{4}+o(1)\right)|G|$, and $\frac{1}{4}$ is best possible. We also consider a related problem for infinite shift graphs.

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Andrii Arman, Vojtěch Rödl, Marcelo Tadeu Sales. 2021-11-22. Independent sets in subgraphs of a shift graph. https://arxiv.org/abs/2105.10971

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