The board / combinatorics / #499
Erdős Problem #499
PROVED LEANdatabase updated 2025-11-29
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(M\) be a real \(n \times n\) doubly stochastic matrix. Does there exist some \(σ \in S_n\) such that \[ \prod_{1 \leq i \leq n} M_{i, σ(i)} \geq n^{-n}? \] This is true, and was proved by Marcus and Minc [MaMi62] [MaMi62] Marcus, Marvin and Minc, Henryk, Some results on doubly stochastic matrices. Proc. Amer. Math. Soc. (1962), 571-579.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/499.Prizenone
Topicscombinatorics
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/499
Attempts
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