Unsolved Problems

For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets?...

What is the maximum size of a cap set in $\mathbb{F}_3^n$?...

What is the exact value of the Ramsey number $R(5,5)$?...

Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions?...

If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\geq 1/k$ from others) at some time?...

For a finite family of sets closed under unions, must some element appear in at least half the sets?...

For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$?...

What is the largest possible cap set in $n$-dimensional affine space over the three-element field?...

If k runners with distinct speeds run on a unit circle, will each runner be "lonely" (≥1/k away from others) at some time?...

Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?...

For any finite union-closed family of sets, does some element appear in at least half the sets?...

What is the exact value of the Ramsey number R(5,5)?...