Extremal number ($\mathrm{ex}(n,H)$)

The extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $n$-vertex graph containing no subgraph isomorphic to $H$.

For fixed positive integers $r,s$ and sufficiently large $n$, the unique extremal graph forbidding $s$ vertex-disjoint copies of the complete graph $K_{r+1}$ is the join of graphs $K_{s-1}+T_r(n-s+1)$, up to isomorphic graphs. Each forbidden clique would require a distinct vertex of $K_{s-1}$.

For every fixed graph $H$, the sequenceis nonincreasing once $n\ge 2$. Indeed, choose a uniformly random $m$-vertex induced subgraph of an extremal $H$-free graph on $n$ vertices. It remains $H$-free and has expected edge countwhich cannot exceed $\mathrm{ex}(m,H)$.

Every triangle-free graph on $n$ vertices has at most $\lfloor n^2/4\rfloor$ edges.

Every graph on $2n$ vertices with at least $n^2+1$ edges contains at least $n$ triangles. An induction deletes the endpoints of an edge lying in no triangle; if every edge lies in a triangle, the edge-triangle incidence bound is already sufficient.