On the asymptotic minimum number of monochromatic 3-term arithmetic progressions

Details On the asymptotic minimum number of monochromatic 3-term arithmetic progressions On the asymptotic minimum number of monochromatic 3-term arithmetic progressions

2006-09-19

arxiv.org/abs/math/0609532v2

Journal of Combinatorial Theory, Series A. Volume 115, Issue 1, January 2008, pp. 185-192.

Mathematics

Combinatorics

9 pages. Revised version fixes formatting errors (same text)

Scientific paper

10.1016/j.jcta.2007.03.006

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.

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