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On the $\ell$-th largest degree of an intersecting family

· 2026 · math.CO · arXiv 2602.01692

2 Pith papers cite this work. Polarity classification is still indexing.

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abstract

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family. For an element $i\in[n]$, the degree of $i$ is the number of sets in $\mathcal{F}$ that contain $i$. Assume that the degrees are ordered as $d_{1}\ge d_{2}\ge\cdots\ge d_{n}$.Huang and Zhao showed that if $n>2k$, then the minimum degree satisfies $d_{n}\le\binom{n-2}{k-2}$, with the maximum attained by the $1$-star. We strengthen this result by proving that for $n\ge 2k+1$, the $(2k+1)$-th largest degree satisfies $d_{2k+1}\le\binom{n-2}{k-2}$, thereby confirming a conjecture of Frankl and Wang. Furthermore, we prove that for large $k$ and $n>12k$, the $(k+2)$-th largest degree $d_{k+2}$ is already at most $\binom{n-2}{k-2}$. The techniques we developed also yield a tight upper bound for the $(\ell+1)$-th largest degree $d_{\ell+1}$ for $\varepsilon k \le \ell \le k$ and sufficiently large $n>C_{\varepsilon} k$.

fields

math.CO 2

years

2026 2

verdicts

UNVERDICTED 2

representative citing papers

On degree bounds of $k$-uniform hypergraphs with bounded matching number

math.CO · 2026-05-20 · unverdicted · novelty 7.0 · 2 refs

Proves that d_{2sk+1} > binom(n-1,k-1) - binom(n-s,k-1) implies matching number at least s in k-uniform hypergraphs (n>2sk), generalizes prior results, shows k+2s-2 is optimal for a relaxed index, and improves the n bound for the Ore-degree condition to 3sk.

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Showing 2 of 2 citing papers.

  • On degree bounds of $k$-uniform hypergraphs with bounded matching number math.CO · 2026-05-20 · unverdicted · none · ref 13 · 2 links · internal anchor

    Proves that d_{2sk+1} > binom(n-1,k-1) - binom(n-s,k-1) implies matching number at least s in k-uniform hypergraphs (n>2sk), generalizes prior results, shows k+2s-2 is optimal for a relaxed index, and improves the n bound for the Ore-degree condition to 3sk.

  • Matchings in hypergraphs via Ore-degree conditions math.CO · 2026-03-06 · unverdicted · none · ref 22 · internal anchor

    New Ore-degree conditions guarantee matchings and s pairwise disjoint edges in r-uniform hypergraphs under size and intersecting assumptions.