Razborov closure

Fix integers $r$ and $l$. The Razborov closure of a family of vertex sets of size at most $l$ repeatedly adjoins a set $W$ whenever there are $r$ existing sets $W_1,\dots ,W_r$ whose pairwise intersections are contained in $W$. A family equal to its closure is called $r$-closed.

An inclusion-minimal $s$-set in an upward-closed family cannot belong to an $r$-member delta-system of such minimal sets: its proper core would be adjoined by Razborov closure, violating minimality. The Erdős–Rado sunflower lemma therefore bounds the number $a_s$ of minimal $s$-sets. This weaker factorial bound is sufficient for the superpolynomial monotone clique lower bound.

An $r$-closed family has at most $(r-1{)}^k$ inclusion-minimal members of cardinality $k$. This limits the number of cliques accepted by a proper closed approximation.