Density threshold for three elements and all their pairwise sums
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How dense must A⊆{1,…,N} be to contain distinct a,b,c such that a+b, a+c and b+c also belong to A? The target is the sharp asymptotic density threshold 5/8.
Why it matters and what counts as progress
Why it matters: A matching threshold separates unavoidable additive structure from constructions avoiding it.
What counts as progress: An upper bound |A|≥5N/8+O(1) forcing the configuration, matched by avoiding sets of asymptotic density 5/8.
How the problem developed
- 1975 · Partial result [1]
Choi, Erdős, and Szemerédi Established earlier threshold results for sets containing three elements and their pairwise sums; these did not identify the sharp asymptotic threshold 5/8.
Ricky Cipollini Proved the sharp asymptotic threshold 5N/8+O(1), matching the standard construction and resolving the three-element case.
AI contribution
A sharp 5/8 bound for an Erdős-Sós pairwise-sums problem
Theorem 1.1 gives a constant C such that every sufficiently large A⊆{1,…,N} with |A|≥5N/8+C contains distinct a,b,c for which a+b, a+c and b+c also lie in A. An interval-union construction shows the 5/8 coefficient is sharp.
Sources
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Publication dates and editorial changes.
Cipollini and GPT establish the sharp pairwise-sums threshold