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

TL;DR

This paper investigates the maximum size of triangle-free vertex sets in the r-distance graph of the hypercube, providing upper bounds for even r and exploring lower bounds across different r regimes.

Contribution

It establishes an upper bound on the size of triangle-free sets for even r and explores lower bounds in various parameter regimes.

Findings

Proves T(n,r)=O((r*2^n)/(n+1)) for even r ≤ n/2.

Provides lower bounds for T(n,r) in different regimes of r.

Advances understanding of combinatorial structures in hypercube graphs.

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 \[T(n,r)=O\!\left(\frac{r2^n}{n+1}\right).\] We also obtain lower bounds in various regimes of $r$ as a function of $n$.