The solution to the problem of finding the minimum number of monochromatic triples $(x,y,x+ay)$ with $a\geq 2$ being a fixed positive integer over any 2-coloring of $[1,n]$ was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky's proof (2003) on the minimum number of monochromatic Schur triples $(x,y,x+y)$. We do this by exploiting the combinatorial nature of the original proof and adapting it to the general problem.
@article{10_37236_6490,
author = {Thotsaporn Thanatipanonda and Elaine Wong},
title = {On the minimum number of monochromatic generalized {Schur} triples},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {2},
doi = {10.37236/6490},
zbl = {1361.05134},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6490/}
}
TY - JOUR
AU - Thotsaporn Thanatipanonda
AU - Elaine Wong
TI - On the minimum number of monochromatic generalized Schur triples
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/6490/
DO - 10.37236/6490
ID - 10_37236_6490
ER -
%0 Journal Article
%A Thotsaporn Thanatipanonda
%A Elaine Wong
%T On the minimum number of monochromatic generalized Schur triples
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/6490/
%R 10.37236/6490
%F 10_37236_6490
Thotsaporn Thanatipanonda; Elaine Wong. On the minimum number of monochromatic generalized Schur triples. The electronic journal of combinatorics, Tome 24 (2017) no. 2. doi: 10.37236/6490
