fact not locally forcing, in stark contrast to the induced graph problem in which Simonovits-Sós conjecture states that all graphs on $4$ or more vertices should be locally forcing. In particular, one can easily find a regular tournament which fails conditions (ii) or (iii) whereas in the induced graph case any regular graph is known to be locally forcing.

This might suggest that there are essentially no non-transitive locally forcing tournaments, as in the global forcing case. We show that this is also not the case and find many non-transitive examples of locally forcing tournaments.

\[thm: exist-locally-forcing\] For any large enough $h$ there exist non-transitive, $h$-vertex, locally forcing tournaments.

Our examples of locally forcing tournaments come from the following random construction. We take a Steiner triple system on $h$ vertices, that is a partition of the edge set of the complete graph on $h$ vertices into edge disjoint triangles. Now we orient each of these triangles into one of two cycles of length three uniformly at random and independently between triangles. We show that this produces with high probability a locally forcing tournament. It is well known (see [@Kirkman]) that Steiner triple systems exist provided $h\equiv 1,3 \pmod{6}$ and our 