The conjecture of van der Waerden, which states that the permanent of a doubly stochastic matrix is at least $n^{-n} n!$.

A weaker version of Erdős' problem 499, which asks whether for every doubly stochastic matrix, there exists a permutation $σ \in S_n$ with $M_{i, σ(i)} ≠ 0$ and such that $$\sum_{1 \leq i \leq n} M_{i, σ(i)} \geq 1$$ Proved by Marcus and Ree [MaRe59].

[MaRe59] Marcus, M. and Ree, R., Diagonals of doubly stochastic matrices. Quart. J. Math. Oxford Ser. (2) (1959), 296-302.