Maximum Number Of Sum-Free Colorings In Finite Abelian Groups

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 with a + b = c. We investigate κr,G, the maximum number of sum-free r-colorings admitted by subsets of G, and our results show a close relationship between κr,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 ≡ 2 (mod 3). For r = 2, 3 we show that a subset A ⊂ G achieves κr,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 κr,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.