Maximum number of sum-free colorings in finite abelian groups

作者：Hiep Han^*; Jimenez Andrea

来源：Israel Journal of Mathematics, 2018, 226(2): 505-534.

DOI：10.1007/s11856-018-1705-1

An r-coloring of a subset A of a finite abelian group G is called sum-free if it does not induce a monochromatic Schur triple, i.e., a triple of elements a, b, c a A with a + b = c. We investigate kappa (r) ,G, the maximum number of sum-free r-colorings admitted by subsets of G, and our results show a close relationship between kappa (r) ,G and largest sum-free sets of G.
Given a sufficiently large abelian group G of type I, i.e., |G| has a prime divisor q with q ae 2 (mod 3). For r = 2, 3 we show that a subset A aS, G achieves kappa (r) ,G if and only if A is a largest sum-free set of G. For even order G the result extends to r = 4, 5, where the phenomenon persists only if G has a unique largest sum-free set. On the contrary, if the largest sum-free set in G is not unique, then A attains kappa (r) ,G if and only if it is the union of two largest sum-free sets (in case r = 4) and the union of three ("independent") largest sum-free sets (in case r = 5).
Our approach relies on the so called container method and can be extended to larger r in case G is of even order and contains sufficiently many largest sum-free sets.

• 出版日期2018-6