The Simonovits-Sós conjecture on 3AP-intersecting families

Partial: any 3AP-intersecting family has size at most $(\tfrac12 - c)2^n$ for an absolute constant $c>0$, the first bound below the trivial one. The conjectured maximum $2^{n-3}$ is not established. The same bound is proved for $H$-intersecting families whenever $H$ is a 3-graph on $[n]$ with bounded codegrees, and a clique shows the codegree assumption cannot be removed.

Partial: any 3AP-intersecting family has size at most $(\tfrac12 - c)2^n$ for an absolute constant $c>0$, the first bound below the trivial one. The conjectured maximum $2^{n-3}$ is not established. The same bound is proved for $H$-intersecting families whenever $H$ is a 3-graph on $[n]$ with bounded codegrees, and a clique shows the codegree assumption cannot be removed.