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arXiv · 2606.20367

On the maximum density of $r$-graphs in which every $(r+1)$-set spans $0$ or $2$ edges

Vishesh Jain·Haoran Luo·Dhruv Mubayi
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Abstract

In 1984, Frankl and F\"uredi asked for the maximum density of an $n$-vertex $r$-graph in which every $(r+1)$-set of vertices spans $0$ or $2$ edges. They gave a construction with asymptotic density $2^{1-r}$. We significantly improve this bound by constructing such $r$-graphs with density $\Omega(r^{-3})$, thereby improving the dependence on $r$ from exponential to polynomial. We also obtain lower bounds for the more general problem in which every $(r+1)$-set spans an even number of edges from $\{0,2,\ldots,2k\}$.

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Vishesh Jain, Haoran Luo, Dhruv Mubayi. 2026-06-18. On the maximum density of $r$-graphs in which every $(r+1)$-set spans $0$ or $2$ edges. https://arxiv.org/abs/2606.20367

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