Sparse hypergraphs with low independence number

TL;DR

This paper constructs a 3-uniform hypergraph with low independence number that avoids a complete 3-uniform hypergraph on 4 vertices, answering a longstanding question and providing counterexamples to related conjectures.

Contribution

It provides the first explicit construction of a K_4-free 3-uniform hypergraph with low independence number, resolving a question from 1981.

Findings

Constructed a K_4-free hypergraph with independence number at most 2N/d^{1/2}.

Provided counterexamples to several conjectures in hypergraph theory.

Improved lower bounds for certain hypergraph Ramsey numbers.

Abstract

Let K_4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai, Erd\H{o}s, Koml\'os, and Szemer\'edi (1981) asked if there is a function \omega(d) tending to infinity such that every 3-uniform, K_4-free hypergraph N vertices and average degree d has independence number at least \omega(d) N/d^{1/2}. We answer this question by constructing a 3-uniform, K_4-free hypergraph with independence number at most 2N/d^{1/2}. We also provide counterexamples to several related conjectures and improve the lower bound of some hypergraph Ramsey numbers.