Sparse hypergraphs with low independence number
Combinatorics
2014-07-24 v4
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.
Keywords
Cite
@article{arxiv.1312.0813,
title = {Sparse hypergraphs with low independence number},
author = {Jeff Cooper and Dhruv Mubayi},
journal= {arXiv preprint arXiv:1312.0813},
year = {2014}
}