Triangle-free subsets of the $r$-distance graph of the hypercube

Abstract

Given the $r$-distance graph on the hypercube $\mathbb{F}_2^n$, where two vertices are adjacent if their Hamming distance is exactly $r$, we study the maximum size $T(n,r)$ of a triangle-free set of vertices. For even $r\le n/2$, we prove \[ T(n,r)=O\!\left(\frac{r2^n}{n+1}\right). \] In particular, $T(n,r)=o(2^n)$ whenever $r=o(n)$. For fixed $0<α<2/3$, we also prove that if $r=αn$, then \[ T(n,r)\le 2^{(1-\varepsilon_α)n} \] for some $\varepsilon_α>0$. We also obtain lower bounds in various regimes of $r$ as a function of $n$.