On independent sets in uncrowded uniform hypergraphs

We prove an average-degree lower bound on the independence number of uncrowded uniform hypergraphs. For every fixed $r\ge 2$ and every $\eta >0$, there exists $d_{\ast }=d_{\ast }(r,\eta )$ such that any uncrowded $(r+1)$-uniform hypergraph $G$ with $n$ vertices and average degree $d\ge d_{\ast }$ satisfies $$\alpha (G)\ge (1-\eta )r^{-1\mathrm{/}r}{\left(\frac{\log d}{d}\right)}^{1\mathrm{/}r}n\mathrm{.}$$ 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 $2,\dots ,r+1$, we track the maximum degrees in every layer. A binomial-type recurrence for this degree profile yields the stated leading constant.