Game saturation of intersecting families

• Autorzy: Balázs Patkós , Máté Vizer
• Języki publikacji: EN

Abstrakty

EN

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, gsatF(IIn,k) and gsatS(IIn,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 $\Omega _k (n^{k/3 - 5} ) \leqslant gsat_F (\mathbb{I}_{n,k} ),gsat_S (\mathbb{I}_{n,k} ) \leqslant O_k (n^{k - \sqrt {k/2} } )$.

Słowa kluczowe

EN

Intersecting families of sets; Saturated families; Positional games

Kategorie tematyczne

• 05D05: Extremal set theory
• 91A24: Positional games (pursuit and evasion, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 9
• Strony: 1382-1389

Daty

wydano

2014-09-01

online

2014-05-08

Twórcy

autor

Balázs Patkós

• Alfréd Rényi Institute of Mathematics

autor

Máté Vizer

• Alfréd Rényi Institute of Mathematics

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-014-0420-3