A note on the ErdősHajnal hypergraph Ramsey problem
Abstract
We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a steppingup lemma established by the first two authors, analogous sharp results are proved for $k$uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 DOI:
 10.48550/arXiv.2003.00074
 arXiv:
 arXiv:2003.00074
 Bibcode:
 2020arXiv200300074M
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 12 pages, 1 figure