Exactly $2^{1012}$ of the subsets of $\{1, 2, \ldots, 2023\}$ are colored red. Is it always true that there exist three distinct red sets $A$, $B$, and $C$ such that every element of $A$ belongs to at least one of $B$ or $C$?
Problem
Source: XII International Festival of Young Mathematicians Sozopol 2023, Theme for 10-12 grade
Tags: combinatorics