Four-sums threshold in Erdős Problem 866

What extra size beyond n forces a subset A⊂{1,…,2n} to contain all six pair sums of four distinct integers? The four integers need not be positive and need not themselves lie in A; the sharp target is n+3 elements for n≥2.

Why it matters and what counts as progress

Why it matters: The exact constant sharpens a finite additive-configuration threshold and requires the correct unrestricted choice of the four integers.

What counts as progress: Prove the threshold and construct a sharp avoiding example showing the constant cannot be lowered.

How the problem developed

1975 · Bound [1]
S. L. G. Choi, Paul Erdős and Endre Szemerédi Established that a constant additive excess above n forces pairwise sums of four distinct integers into a subset of [1,2n], without determining the optimal constant.
28 Apr 2026 · Reported theorem [2]
Wouter van Doorn Proved that excess 3 suffices and is optimal for four distinct positive integers; for the version allowing a non-positive integer, the threshold is 3 for n≥2.

AI contribution

The cardinality of a set containing the pairwise sums of a fixed number of integers

Wouter van Doorn

For every n≥2, g₄(n)=3: any A⊆{1,…,2n} with at least n+3 elements contains all six pairwise sums of four distinct integers, and the constant cannot be lowered. The four integers need not be positive or members of A.

AI involvementAI-autonomous

ImportanceModest

Technical verificationFormal (Lean 4 via Aristotle)

Human understandingFull