Great problems

20th century

1. 1932
   Erdős discrepancy problemCombinatorics and discrete mathematics, Arithmetic and number theory. Every infinite $\pm 1$ sequence has arbitrarily large partial sums along some homogeneous arithmetic progression $d, 2d, \dots, nd$; proved by Terence Tao in 2015.ErdősTTTerence Tao, 2015
2. 1972
   Erdős–Faber–Lovász conjectureCombinatorics and discrete mathematics. Can the union of $n$ cliques of size $n$ that pairwise share at most one vertex always be properly vertex-colored with $n$ colors? Posed by Paul Erdős, Vance Faber, and László Lovász in 1972, the conjecture was proved for all sufficiently large $n$ in 2021 by Kang, Kelly, Kühn, Methuku, and Osthus (published in the Annals of Mathematics in 2023), while small cases $n \le 12$ are verified computationally and intermediate values remain open.ErdősSolved