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arXiv · 2506.18782

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

Padmini Mukkamala·Ananthakrishnan Ravi
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Abstract

Given the $r$-distance graph on the hypercube $\F_2^n$, 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 $0<\alpha<2/3$, we also prove that if $r=\alpha n$, where $n$ ranges over integers such that $\alpha n$ is an even integer, then \[ T(n,r)\le 2^{(1-\varepsilon_\alpha)n} \] for some $\varepsilon_\alpha>0$. We also obtain lower bounds in various regimes of $r$ as a function of $n$.

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Padmini Mukkamala, Ananthakrishnan Ravi. 2025-06-23. Triangle-free subsets of the $r$-distance graph of the hypercube. https://arxiv.org/abs/2506.18782

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