Abstract
We construct an $S_{3}$ -symmetric probability distribution on $\{(a,b,c)\in \mathbb{Z}_{{\geqslant}0}^{3}\,:\,a+b+c=n\}$ such that its marginal achieves the maximum entropy among all probability distributions on $\{0,1,\ldots ,n\}$ with mean $n/3$ . Existence of such a distribution verifies a conjecture of Kleinberg et al. [‘The growth rate of tri-colored sum-free sets’, Discrete Anal. (2018), Paper No. 12, arXiv:1607.00047v1], which is motivated by the study of sum-free sets.
Highlights
The recent breakthrough by Croot, Lev and Pach [2] and the subsequent solution of the cap-set problem by Ellenberg and Gijswijt [3] led to a dramatic improvement of known upper bounds on the size of maximum sum-free sets in powers of finite groups
A tri-colored sum-free set in Fnp is a collection of triples {(xi, yi, zi )}im=1 of elements of Fnp such that xi + y j + zk = 0 if and only if i = j = k
In the remainder of the section, we describe a collection of such shifts which transform β into a nonnegative vector
Summary
The recent breakthrough by Croot, Lev and Pach [2] and the subsequent solution of the cap-set problem by Ellenberg and Gijswijt [3] led to a dramatic improvement of known upper bounds on the size of maximum sum-free sets in powers of finite groups. Sawin and Speyer establish an upper bound m eγpn on the size of a tri-colored sum-free set in Fnp, where γp is as follows. (In the published version of [5], the upper bound part of the statement of Theorem 1, as well as the examples for n 25 referenced later, were removed, as the proofs of Theorem 2 in an earlier version of this article, and independent work of Pebody [6], showed that they were unnecessary, but they are still available in the referenced arxiv version.) All tri-colored sum-free sets in Fnp have size at most eγpn. Examples of the distributions satisfying Theorem 2 for n 25 are provided in [5] Based on these examples and additional experimentation, we construct a simple S3-symmetric function of {(a, b, c) ∈ Z3 0 : a + b + c = n} with the marginal given by (1).
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