Erdős–Moser sum-free set extremal function

Determine the largest possible cardinality of a subset B of a finite non-empty set A of integers such that b_1+b_2 does not belong to A for every pair of distinct elements b_1,b_2 in B, and characterize the resulting extremal function φ(n) obtained by minimizing this cardinality over all n-element integer sets.

Background

The paper introduces the Erdős–Moser problem for a finite non-empty set A of integers. A subset B of A is sum-free with respect to A when no sum of two distinct elements of B lies in A, and M(A) denotes the largest size of such a subset. The function φ(n) is defined as the minimum of M(A) over all integer sets A of size n. The paper reviews substantial progress on lower and upper bounds for φ(n), but the extremal question itself remains unresolved.

References

The motivation for this paper comes from the following old question of Erdős and Moser p. 187, which was recently reiterated by Green in his list of open problems Problem 2 (see also Problem #787). Given a finite non-empty set of integers $A$, what is the largest size of a subset $B \subseteq A$ which is sum-free with respect to $A$?

— The Erdős--Moser sum-free set problem via improved bounds for $k$-configurations  (2501.10203 - Beker, 17 Jan 2025) in Section 1, Introduction