Independent Sets in Subgraphs of a Shift Graph

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