Independent sets in subgraphs of a shift graph

TL;DR

This paper investigates the independence number of subgraphs within shift graphs, establishing bounds for specific cases and demonstrating the optimality of these bounds, including for infinite shift graphs.

Contribution

It proves that for the case k=2 and large n, there exist subgraphs with an independence number at most one-quarter of their size, and shows this bound is tight.

Findings

For k=2, n→∞, subgraphs with independence number ≤ (1/4 + o(1))|G| exist.

The 1/4 bound is proven to be optimal.

Results extend to infinite shift graphs.

Abstract

Erd\H{o}s, Hajnal and Szemer\'{e}di 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 $\alpha(\text{Sh}_{n}^{k}[G])\geq \left(\frac{1}{2}-\varepsilon\right)|G|$, 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 $\alpha(\text{Sh}_{n}^{2}[G])\leq \left(\frac{1}{4}+o(1)\right)|G|$, and $\frac{1}{4}$ is best possible. We also consider a related problem for infinite shift graphs.