ON RAMSEY-TURAN NUMBERS FOR 3-GRAPHS

For every r-graph G let pi(G) be the minimal real number-pi such that for every epsilon > 0 and n > n0(epsilon, G) every r-graph H with n vertices and more than (pi + epsilon)(r(n)) edges contains a copy of G. The real number lambda(G) is defined in the same way, adding the constraint that all independent sets of vertices in H have size o(n). Erdos and Sos asked whether there exist r-graphs G with pi(G) > lambda(G) > 0. Frankl and Rodl proved that there exist infinitely many such r-graphs for every r greater-than-or-equal-to 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.