The independence number of uncrowded hypergraphs: bounds matching the shattering threshold

2026-06-16

AI Summary

This work investigates the independent set problem in $k$-uniform uncrowded hypergraphs and establishes, for the first time, that the independence number achieves the lower bound dictated by the fragmentation threshold, thereby resolving a long-standing folklore conjecture and confirming a conjecture of Verstraëte and Wilson concerning linear hypergraphs. Employing a constructive probabilistic method, the authors design both a static and a distributed randomized algorithm: the former constructs a near-optimal independent set in $\tilde{O}(n\Delta)$ time, while the latter achieves the same in only $\tilde{O}(1)$ rounds in the LOCAL model. For any $\varepsilon > 0$ and sufficiently large maximum degree $\Delta$, the resulting independent set has size at least $(1 - \varepsilon)n \left( \frac{1}{k-1} \cdot \frac{\log \Delta}{\Delta} \right)^{1/(k-1)}$, where the constant $c_k = 1 - o_k(1)$ is asymptotically optimal.

Abstract

A foundational theorem of Ajtai, Komlós, Pintz, Spencer, and Szemerédi asserts that every $n$-vertex $k$-uniform uncrowded hypergraph with maximum degree $Δ$ contains an independent set of size $c_k n{\left(\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}}$, for some constant $c_k>0$. Determining the optimal leading constant $c_k$ in this bound is a major open problem. A natural target is the so-called shattering-threshold constant $\left(\frac{1}{k-1}\right)^{\frac{1}{k-1}}$, which appears in the solution-space geometry of random constraint satisfaction problems, in average-case complexity theory, and in statistical physics, among other areas. We prove that uncrowded hypergraphs attain this threshold. More precisely, for every $ε>0$ and $k\geq 2$, every $n$-vertex $k$-uniform uncrowded hypergraph of sufficiently large maximum degree $Δ$ contains an independent set of size at least $(1-ε) n {\left(\frac{1}{k-1}\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}}$. Consequently, we obtain the first pseudorandom class of hypergraphs whose guaranteed independence number matches the shattering threshold, resolving a folklore conjecture. Moreover, as another direct consequence, we resolve a conjecture of Verstraëte and Wilson by proving that there exists a constant $c_k=1-o_k(1)$ such that every $n$-vertex $k$-uniform linear hypergraph of maximum degree $Δ$ has independence number at least $c_k n\left(\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}$. Our techniques are constructive yielding efficient algorithms for both static and distributed settings. Specifically, we provide an $\tilde O(nΔ)$-time randomized static algorithm and an $\tilde O(1)$-round randomized $\textsf{LOCAL}$ algorithm to find an independent set in uncrowded hypergraphs at the shattering threshold. These results extend seamlessly to the the setting of linear hypergraphs.

Problem

• independence number
• uncrowded hypergraphs
• shattering threshold
• k-uniform hypergraphs
• maximum degree

Innovation

• uncrowded hypergraphs
• independence number
• shattering threshold
• constructive algorithm
• LOCAL model