All Random Solved Random Open
Let $M=(a_{ij})$ be a real $n\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there exist some $\sigma\in S_n$ such that \[\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}?\]
A weaker form of the conjecture of van der Waerden, which states that \[\mathrm{perm}(M)=\sum_{\sigma\in S_n}\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}n!\] with equality if and only if $a_{ij}=1/n$ for all $i,j$.

This conjecture is true, and was proved by Marcus and Minc [MaMi62].

Erdős also conjectured the even weaker fact that there exists some $\sigma\in S_n$ such that $a_{i\sigma(i)}\neq 0$ for all $i$ and \[\sum_{i}a_{i\sigma(i)}\geq 1.\] This weaker statement was proved by Marcus and Ree [MaRe59].

van der Waerden's conjecture itself was proved by Gyires [Gy80], Egorychev [Eg81], and Falikman [Fa81].