Existence of a target graph with no bounded-move winning strategy on a single complete board

Determine whether there exists a graph H such that Player 1 does not have a winning strategy in the strong Ramsey game R(K_n,H) within a number of moves bounded independently of n as n tends to infinity.

Background

The strong Ramsey game R(K_n,H) is played on the complete graph K_n, with Player 1 and Player 2 alternately claiming edges and Player 1 seeking to claim a copy of H. Ramsey theory guarantees that Player 1 eventually wins for sufficiently large n, but this does not establish that the number of moves needed can be bounded solely in terms of H.

The paper studies the two-board variant R(K_n \sqcup K_n,H) and constructs infinitely many target graphs for which Player 1 cannot win within a bounded number of moves. The authors present this as evidence relevant to, but not a resolution of, the corresponding longstanding single-board problem.