Game saturation of intersecting families

We consider the following combinatorial game: two players, Fast and Slow, claim k-element subsets of [n] = {1, 2, …, n} alternately, one at each turn, so that both players are allowed to pick sets that intersect all previously claimed subsets. The game ends when there does not exist any unclaimed k-subset that meets all already claimed sets. The score of the game is the number of sets claimed by the two players, the aim of Fast is to keep the score as low as possible, while the aim of Slow is to postpone the game’s end as long as possible. The game saturation numbers, gsat_F(II _n,k ) and gsat_S(II _n,k ), are the score of the game when both players play according to an optimal strategy in the cases when the game starts with Fast’s or Slow’s move, respectively. We prove that $Omega _k (n^{k/3 - 5} ) leqslant gsat_F (mathbb{I}_{n,k} ),gsat_S (mathbb{I}_{n,k} ) leqslant O_k (n^{k - sqrt {k/2} } )$ .