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,
such 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 number is the score of the game when both
players play according to an optimal strategy. To be precise we have to
distinguish two cases depending on which player takes the first move. Let
gsatF(In,k) and gsatS(In,k) denote the score of
the saturation game when both players play according to an optimal strategy and
the game starts with Fast's or Slow's move, respectively. We prove that
Ωk(nk/3−5)≤gsatF(In,k),gsatS(In,k)≤Ok(nk−k/2) holds