On the order of intersecting hypergraphs

TL;DR

This work develops order-analogues for classical extremal results on intersecting hypergraphs, focusing on the parameter Ord$(\mathcal{H})$ (the number of non-isolated vertices). It establishes sharp-order bounds in several regimes: with bounded maximum degree via a kernel-based reduction leading to $\mathrm{Ord}(\mathcal{H}) \le \frac{1}{4}k^{2}\Delta + \frac{3}{2}\binom{2k-2}{k-1}$, for λ-intersecting graphs with asymptotics $\mathrm{Ord}(\mathcal{H}) \le \frac{4}{27}(k-\lambda)^{3} + o(k^{3})$ when $\lambda = o(k)$, and for related set-pair systems via Bollobás-type bounds. The paper also provides tight constructions (notably using projective planes) that nearly achieve these bounds and derives a complete order-bound for $1$-cross-intersecting SPS, with exact asymptotics when bounds align with divisibility conditions. Overall, the results connect kernel structure, intersection constraints, and extremal order, offering insight into how order behaves under classical Higman-type constraints and highlighting several open problems in the area.

Abstract

Determining the maximum number of edges in an intersecting hypergraph on a fixed ground set under additional constraints is one of the central topics in extremal combinatorics. In contrast, there are few results on analogous problems concerning the maximum order of such hypergraphs. In this paper, we systematically study these vertex analogues.

Theorems & Definitions (32)

• Theorem 1.1: Füredi Furedi
• Theorem 1.2: Deza Deza
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: FGK23
• Theorem 1.6
• Definition 2.1
• Theorem 2.2: Majumder Majumder
• proof : Proof of \ref{['thm:furedi-order']}
• Proposition 2.3