3 research outputs found
Monochromatic 4-term arithmetic progressions in 2-colorings of
This paper is motivated by a recent result of Wolf \cite{wolf} on the minimum
number of monochromatic 4-term arithmetic progressions(4-APs, for short) in
, where is a prime number. Wolf proved that there is a 2-coloring of
with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the
proof is probabilistic and non-constructive. In this paper, we present an
explicit and simple construction of a 2-coloring with 9.3% fewer monochromatic
4-APs than random 2-colorings. This problem leads us to consider the minimum
number of monochromatic 4-APs in for general . We obtain both lower
bound and upper bound on the minimum number of monochromatic 4-APs in all
2-colorings of . Wolf proved that any 2-coloring of has at least
monochromatic 4-APs. We improve this lower bound into
.
Our results on naturally apply to the similar problem on (i.e.,
). In 2008, Parillo, Robertson, and Saracino \cite{prs}
constructed a 2-coloring of with 14.6% fewer monochromatic 3-APs than
random 2-colorings. In 2010, Butler, Costello, and Graham \cite{BCG} extended
their methods and used an extensive computer search to construct a 2-coloring
of with 17.35% fewer monochromatic 4-APs (and 26.8% fewer monochromatic
5-APs) than random 2-colorings. Our construction gives a 2-coloring of
with 33.33% fewer monochromatic 4-APs (and 57.89% fewer monochromatic 5-APs)
than random 2-colorings.Comment: 23 pages, 4 figure
Exact Lower Bounds for Monochromatic Schur Triples and Generalizations
We derive exact and sharp lower bounds for the number of monochromatic
generalized Schur triples whose entries are from the set
, subject to a coloring with two different colors. Previously,
only asymptotic formulas for such bounds were known, and only for
. Using symbolic computation techniques, these results are
extended here to arbitrary . Furthermore, we give exact
formulas for the minimum number of monochromatic Schur triples for ,
and briefly discuss the case .Comment: 24 Page
On the monochromatic Schur Triples type problem
We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of [1, n], of monochromatic {x, y, x + ay} triples for a β₯ 1. We give a new proof of the original case of a = 1. We show that the minimum number + O(n) when a β₯ 2. We also find a new upper bound for the minimum number, over all r-colorings of [1, n], of monochromatic Schur triples, for r β₯ 3. of such triples is at most n 2 2a(a 2 +2a+3)