The number of additive triples in subsets of abelian groups

Abstract

A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a set $A$ of elements of an abelian group $G$ has cardinality $a$. How many Schur triples must $A$ contain? Moreover, which sets of $a$ elements of $G$ have the smallest number of Schur triples? In this paper, we answer these questions for various groups $G$ and ranges of $a$.

20 pages; corrected the erroneous equality in (1) in the statement of Theorem 1.3

References in corpus (2)

• Minimum number of monotone subsequences of length 4 in permutations
• On the number of monotone sequences

Cited by in corpus (4)

• Towards a characterisation of Sidorenko systems
• Additive Combinatorics: A Menu of Research Problems
• The Maximum Size of $(k,l)$-Sum-Free Sets in Cyclic Groups
• The largest projective cube-free subsets of ${\mathbb{Z}}_{2^n}$