An extremal problem on Hilbert cubes and complete r-partite hypergraphs
Combinatorics
2013-11-27 v2
Abstract
We construct a set of positive integers A in {1,..., n} with |A|>> n^{2/3} that does not contain Hilbert cubes of dimension 3. As a consequence we prove that ex(n; K^(3)(2,2,2))>> n^{8/3} where K^(3)(2,2,2) is the simplest complete 3-partite hypergraph. This is the first case of an improvement on the trivial lower bound for ex(n; L) when L is a complete r-partite hypergraph.
Keywords
Cite
@article{arxiv.1311.0719,
title = {An extremal problem on Hilbert cubes and complete r-partite hypergraphs},
author = {Javier Cilleruelo},
journal= {arXiv preprint arXiv:1311.0719},
year = {2013}
}
Comments
This paper has been witdrawn by the author because Z. Furedi has communicated us that the same result, although with a distinct constructions, was obtained by Katz, Nets Hawk; Krop, Elliot; Maggioni, Mauro "Remarks on the box problem". Math. Res. Lett. 9 (2002), no. 4, 515--519