SIR Mathematics Investigation Abstract
COLORING FINITE ABELIAN GROUPS TO AVOID MONOCHROMATIC TRIPLES WHOSE SUM IS 0
Presenter:
Abhi Gulati, Illinois Mathematics and Science Academy, 1500 West Sullivan Road, Aurora, IL, 60506; agulati@imsa.edu
Mentor:
Professor László Babai Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637, 773-702-3486; 773-702-8487; laci@cs.uchicago.edu
Abstract:
In this paper, we consider a Ramsey-type question on finite Abelian groups. We wish to color the finite Abelian group G in such a way that for all triples of distinct elements x,y,z in G of the same color, x+y+z is not 0. Let \chi(G), the chromatic number of G, be the minimum number of colors permitting such a coloring. We give a full characterization of those classes of finite Abelian groups G which have the property \chi(G) is bounded; it turns out that for a class F of finite Abelian groups, \chi(G) is bounded if and only if both the 2=ranks and the 3=ranks of the groups in F are bounded. We give explicit lower and upper bounds on \chi(G) in terms of the 2-rank and the 3-rank of G.