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arXiv · 2606.18171

On independent sets in uncrowded uniform hypergraphs

Jing Yu·Junchi Zhang
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Abstract

We prove an average-degree lower bound on the independence number of uncrowded uniform hypergraphs. For every fixed $r\geq 2$ and every $\eta>0$, there exists $d_*=d_*(r,\eta)$ such that any uncrowded $(r+1)$-uniform hypergraph $G$ with $n$ vertices and average degree $d\geq d_*$ satisfies \[ \alpha(G)\geq (1-\eta)r^{-1/r}\left(\frac{\log d}{d}\right)^{1/r}n. \] The proof combines a cleaning procedure, which reduces the maximum top-layer degree to the average scale, with a random nibble procedure that repeatedly extracts independent vertices while controlling all lower-order degrees created by the process. After an initial top-layer cleaning, we run a trace nibble. Since the residual hypergraph contains traces of all sizes $2,\ldots,r+1$, we track the maximum degrees in every layer. A binomial-type recurrence for this degree profile yields the stated leading constant.

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Jing Yu, Junchi Zhang. 2026-06-16. On independent sets in uncrowded uniform hypergraphs. https://arxiv.org/abs/2606.18171

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