Math Talk

[Lino 'Pi' Demasi]

Waterloo graduate Richard Hoshino will be giving a talk entitled "On
Fractional Ramsey Numbers" this Friday, Novemnber 21st from 4:00 - 5:00 in
MC 2034. This talk is being organized by the Pure Math, Applied Math,
Combinatorics & Optimization club. Refreshments will be served and an
abstract is included below. Any questions can be directed to Lino Demasi
at lmde...@student.math.uwaterloo.ca.

Lino Demasi
PMAMC&OC VPPM

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The Ramsey number R(x,y) is defined to be the smallest positive integer n
such that if we colour the edges of the complete graph K_n with two
colours (red and blue), there must either be a red clique of size x, or a
blue clique of size y. We can easily extend this idea and define the
generalized Ramsey number R(a_1, a_2, ..., a_k). Finding an explicit
formula for R(a_1, a_2, ..., a_k) is one of the great unsolved problems of
discrete mathematics, and no one is anywhere close to a solution. In
fact, the only known Ramsey numbers are R(3,3), R(3,4), R(3,5), R(3,6),
R(3,7), R(3,8), R(3,9), R(4,4), R(4,5), and R(3,3,3).

We will start by giving a brief overview of Ramsey theory and define
concepts from fractional graph theory. This enables us to define the
generalized *fractional* Ramsey number R_f(a_1, a_2, ..., a_k). Jacobson,
Levin, and Scheinerman solved the problem for the case k=2 by finding an
explicit formula for R_f(a_1,a_2).

In this talk, we will provide a new result by solving the problem for the
case k=3. We will conjecture an explicit formula for an arbitrary k, and
conclude the talk by suggesting ideas and techniques that may lead to a
complete solution.