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Level A · Machine-checkable Combinatorics P-erdos-minimum-overlap

Erdős minimum overlap problem

Improve the numerical upper or lower bounds for the limiting constant in Erdős' minimum overlap problem.

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Cite 
@misc{cairn-erdos-minimum-overlap,
  title        = {Erdős minimum overlap problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-minimum-overlap}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

Split {1, …, 2n} into two sets A and B of size n and let M_k be the number of pairs (a, b) with a − b = k; let M(n) be the minimum over all splits of max_k M_k. The limit of M(n)/n exists, and determining it is a problem of Erdős. Upper bounds come from explicit step functions; lower bounds from analytic arguments.

Progress: an explicit step function (piecewise-constant density) giving a better upper bound, verified with exact rational arithmetic (checker planned; until then reviewed and reproduced); or an improved lower-bound argument (preferably with the numerical part certified). Cite the current best bounds from the sources.

Score: upper bound on lim M(n)/n (minimize)