On the asymptotic minimum number of monochromatic 3-term arithmetic progressions
Combinatorics
2007-12-18 v2 Optimization and Control
Abstract
Let V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of {1,2,...,n}. We show that (1675/32768) n^2 (1+o(1)) <= V(n) <= (117/2192) n^2(1+o(1)). As a consequence, we find that V(n) is strictly greater than the corresponding number for Schur triples (which is (1/22) n^2 (1+o(1)). Additionally, we disprove the conjecture that V(n) = (1/16) n^2(1+o(1)), as well as a more general conjecture.
Cite
@article{arxiv.math/0609532,
title = {On the asymptotic minimum number of monochromatic 3-term arithmetic progressions},
author = {Pablo A. Parrilo and Aaron Robertson and Dan Saracino},
journal= {arXiv preprint arXiv:math/0609532},
year = {2007}
}
Comments
9 pages. Revised version fixes formatting errors (same text)