There exist caps which block all spaces of fixed codimension in \(\mathbb{P}\mathbb{G} (n,2)\) (Q1906127)

In this 2-page note, the 3 authors prove that for any \(k \geq 0\), and for every \(n\) such that there exists a triangle free graph with \(n + 1\) vertices and of chromatic number larger than \(2^k\), there exists a cap in \(PG(n,2)\) which intersects all subspaces of dimension \(n - k\) of \(PG(n,2)\).