Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.)
For instance, in an n-element set, what is the largest number of k-element subsets that can pairwise intersect one another?
The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory.
Ramsey theory shows that at most five persons can attend such a party (see Theorem on Friends and Strangers).
It appears that (independent of what the given integers actually are) we can always mark at least one-third of them.