An easy upper bound for Ramsey numbers
Abstract
It is observed that the conjugacy growth series of the infinite finitary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. Numerical evidences indicate that the coefficients of a series related to the finitary alternating group seem to satisfy congruence relations reminiscent of Ramanujan's congruences for the partition function.
Cite
@article{arxiv.1603.05243,
title = {An easy upper bound for Ramsey numbers},
author = {Roland Bacher},
journal= {arXiv preprint arXiv:1603.05243},
year = {2016}
}
Comments
This paper contains no new results but presents a slight simplification of the standard proof by Erdos-Szekeres for Ramsey numbers. The main tool is a new (to my knowledge) variation of Ramsey numbers which is perhaps of independent interest