Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.01915

Quadratic bounds for intersecting hypergraphs: An affirmative answer to a question of Duke and Erdős

Guoqing Wang·Linlin Wang·Jun Yue
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $m(k,b)$ denote the minimum number of edges in an intersecting $k$-uniform hypergraph in which every $(k-1)$-set of vertices is disjoint from at least $b$ edges, with $m(k,b)=+\infty$ if no such hypergraph exists. Duke and Erdős [Congressus Numerantium (1980)] asked whether $m(k,k)=o(k^4)$ is possible. In this paper, we answer this question affirmatively and determine the order of growth of $m(k,k)$ by proving that $k^2+1\leq m(k,k)\leq Ck^2$ for all sufficiently large $k$, where $C$ is an absolute constant. More generally, we show that $m(k,b)=Θ(kb)$ for all sufficiently large odd $k$ and every integer $b$ with $k\leq b\leq (k-1)(k-3)/2$, where the implied constants are independent of $k$ and $b$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Guoqing Wang, Linlin Wang, Jun Yue. 2026-11-01. Quadratic bounds for intersecting hypergraphs: An affirmative answer to a question of Duke and Erdős. https://arxiv.org/abs/2611.01915

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Connected Dominating Sets in Triangulations

A dominating set of a graph $G$ is \emph{connected} if it induces a connected graph in $G$. For planar triangulations, it has been known since 1990 that every $n$-vertex triangulation with $n\ge4$ admits a connected dominating set of size at most $n/2 - 1$, and no improvement to this bound was known for over three decades. We break this longstanding barrier by showing that every $n$-vertex triangulation has a connected dominating set of size at most $10n/21$. Equivalently, every triangulation admits a spanning tree with at least $11n/21$ leaves. This result has graph drawing applications, establishing a bound for one-bend free sets and improving the known bound for simultaneous planar embeddings. The connected dominating set results extend readily to show that an $n$-vertex triangulation of a surface of Euler genus $g$ has a connected dominating set of size at most $10n/21+O(\sqrt{gn})$.

math.CO↗
arXiv2026

Curves on the torus with few intersections

Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of $k+O(\sqrt{k}\log k)$. We determine the maximum size of such a set for every k. In particular, the maximum never exceeds k+6, and it does not exceed k+4 when k is large. As this quantity coincides with the maximal number of columns of a generic k-modular matrix with two rows, our result also settles the column number problem, a problem of interest in combinatorial optimization, for such matrices.

math.CO↗
arXiv2026

Spectral extrema on closed surfaces of fixed Euler genus

We investigate a classical extremal problem in combinatorics and topological graph theory: determining the maximum spectral radius $spex(n,γ)$ of all $n$-vertex graphs embeddable on a closed surface of Euler genus $γ$. Building on seminal contributions by Boots and Royle, Cao and Vince, and Hong in the 1990s; Ellingham and Zha in 2000; Dvořák and Mohar in 2010; Tait and Tobin in 2017; and Liu, Ning, and Wang in 2026, we provide a complete characterization of spectral extremal graphs for any fixed Euler genus $γ$ and all sufficiently large $n$. The extremal graph attaining $\operatorname{spex}(n,γ)$ is unique, necessarily edge-extremal, and, roughly speaking, is constructed from the join $K_2 \nabla P_{n-2}$ by adding exactly $3γ$ edges. As a consequence, we derive asymptotically tight estimates for $\operatorname{spex}(n,γ)$, strengthening a classical bound due to Ellingham and Zha [J. Combin. Theory B, 2000]. We further characterize the extremal structures for the projective plane and torus, sharpening earlier bounds of Hong [J. Combin. Theory B, 1995]. For the planar case, we establish an explicit vertex threshold of $4.5\times 10^6$ for the Tait--Tobin theorem. Our approach combines extremal constructions of cellular embeddings, topological edge-switching techniques, and walk-counting estimates for large-scale substructures, revealing a bridge between spectral extremality and surface topology.

math.CO↗