Open problems in Combinatorics

Every finite poset that is not a chain contains two elements x, y such that the fraction of linear extensions with x below y lies between 1/3 and 2/3.

D(n) counts the antichains of subsets of an n-element set — equivalently, monotone Boolean functions on n variables. Compute the next value.

Erdős–Rado: any family of more than C(r)^w sets, each of size w, contains an r-sunflower (r sets with identical pairwise intersections). The conjecture puts C(r) independent of w.

What is the shortest string over n symbols containing every permutation of them as a consecutive substring? Known exactly only for n ≤ 5.

W(r, k) is the smallest N such that every r-coloring of 1…N contains a monochromatic k-term arithmetic progression. Only a handful of values are known exactly.