Journal of Combinatorial Theory Series B· 2026Q1
Two Erdős–Hajnal-type theorems for forbidden order-size pairs
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- 2026year
Short summary
Two new theorems extend the Erdős–Hajnal conjecture to hypergraphs, showing that hypergraphs avoiding specific edge counts within vertex subsets must contain large homogeneous sets or exhibit a wide range of edge counts.
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Key points
- Introduces two hypergraph analogs of the Erdős–Hajnal conjecture.
- Shows that $r$-graphs avoiding a specific number of edges in $m$-vertex subsets have large homogeneous sets.
- Demonstrates that $3$-graphs without polynomial-sized homogeneous sets must have $m$ vertices spanning $\Omega(m^3)$ different edge counts.
AI-generated from the title and abstract; the full text is not read.
Abstract
The celebrated Erdős-Hajnal conjecture says that any graph without a fixed induced subgraph $H$ contains a very large homogeneous set. A direct analog of this conjecture is not true for hypergraphs. In this paper we present two natural variants of this problem which do hold for hypergraphs. We show that for every $r \geq 3$, $m \geq m_0(r)$ and $0 \leq f \leq \binom{m}{r}$, if an $r$-graph $G$ does not contain $m$ vertices spanning exactly $f$ edges, then $G$ contains much larger homogeneous sets than what is guaranteed to exist in general $r$-graphs. We also prove that if a $3$-graph $G$ does not contain homogeneous sets of polynomial size, then for every $m \geq 3$ there are $Ω(m^3)$ values of $f$ such that $G$ contains $m$ vertices spanning exactly $f$ edges. This makes progress on a problem of Axenovich, Bradač, Gishboliner, Mubayi and Weber.
The authors' abstract, as published at the source. Journal of Combinatorial Theory Series B, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics