On ramsey‐tuŕan numbers for 3‐graphs

Abstract For every r ‐graph G let π( G ) be the minimal real number ϵ such that for every ϵ < 0 and n ϵ n 0 (λ, G ) every R ‐graph H with n vertices and more than (π + ϵ)(nr) edges contains a copy of G . The real number λ( G ) is defined in the same way, adding the constraint that all independent sets of vertices in H have size 0( n ). Erdös and Sós asked whether there exist r ‐graphs G with π( G ) < λ( G ) < 0. Frankl and Rödl proved that there exist infinitely many such r ‐graphs for every r ≦ 3. However, no example of an r ‐graph with above property was known. We construct an example of such a 3‐graph with 7 vertices and 9 edges.