Maximum number of sum-free colorings in finite abelian groups

Abstract

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.We are interested in the questions as how large κ r,G can be, given r ≥ 2 and G, and which subsets of G achieve the maximum.A straightforward lower bound for κ r,G is obtained by considering a largest sum-free set B ⊂ G, which gives κ r,G ≥ r |B| . The size of largest sum-free sets of G, denoted by µ(G), is

On the Maximum Number of Integer Colourings with Forbidden Monochromatic Sums

“…During the preparation of this paper, we became aware of the results of Hàn and Jiménez [18] who recently studied similar questions in the setting of finite abelian groups. Given a finite abelian group (Γ, +), define an r-colouring of A ⊆ Γ to be valid if it has no monochromatic sum.…”

“…The results of Hàn and Jiménez show a close relationship between f (Γ, r) and the largest sum-free sets of Γ, and characterise for r 5 the extremal sets. As in [18], our proof begins with an application of the container method. But, in the setting of abelian groups, there is much more structure than in the integer setting, and thus one can obtain much stronger stability, leading to precise results for larger values of r than those obtained here.…”

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“…The Erdős-Rothschild extension for sum-free sets has been pursued by Liu, Sharifzadeh and Staden [29] for subsets of the integers, and Hàn and Jiménez [15] for finite abelian groups. More specifically, they investigated the extremal configurations which maximize the number of sum-free r-colorings.…”

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“…Hoppen, Kohayakawa and Lefmann [19] studied the Erdős-Rothschild extension of the celebrated Erdős-Ko-Rado theorem. Liu, Sharifzadeh and Staden [23] and Hàn and Jiménez [18] determined the maximum number of monochromatic sum-free colorings of integers and of finite abelian groups, respectively. Recently, motivated by these results, Cheng et al [12] studied the number of rainbow sum-free colorings of integers and their typical structure.…”