Abstract:We prove an average-degree lower bound on the independence number of uncrowded uniform hypergraphs. For every fixed integer $r\geq 2$ and every $\eta>0$, there exists $d_*=d_*(r,\eta)$ such that for every $d\geq d_*$, any uncrowded $(r+1)$-uniform hypergraph $G$ with $n$ vertices and average degree $d$ 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 $r$-degree to the average scale, with a random nibble 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.
From: Jing Yu [view email]
[v1]
Tue, 16 Jun 2026 17:07:21 UTC (22 KB)
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