The threshold for the Maker-Breaker H -game Miloˇ s Stojakovi´ c Department of Mathematics and Informatics, University of Novi Sad Joint work with Rajko Nenadov and Angelika Steger. 1 / 15
Introduction 2 / 15
Introduction A positional game : ◮ The board – a finite set X , ◮ the winning sets – F ⊆ 2 X , a collection of subsets of X . ◮ ( X , F ) – the hypergraph of the game. 2 / 15
Introduction A positional game : ◮ The board – a finite set X , ◮ the winning sets – F ⊆ 2 X , a collection of subsets of X . ◮ ( X , F ) – the hypergraph of the game. A Maker-Breaker positional game: ◮ Played by two players - Maker and Breaker , ◮ Maker and Breaker alternately claim unclaimed elements of X , ◮ Maker wins if he claims all elements of some F ∈ F ; otherwise Breaker wins . 2 / 15
Introduction A positional game : ◮ The board – a finite set X , ◮ the winning sets – F ⊆ 2 X , a collection of subsets of X . ◮ ( X , F ) – the hypergraph of the game. A Maker-Breaker positional game: ◮ Played by two players - Maker and Breaker , ◮ Maker and Breaker alternately claim unclaimed elements of X , ◮ Maker wins if he claims all elements of some F ∈ F ; otherwise Breaker wins . A Maker-Breaker positional game on the complete graph : ◮ The board is the edge set of the complete graph K n , ◮ the winning sets are usually representatives of a graph-theoretic structure. 2 / 15
Example 3 / 15
Example Maker-Breaker triangle game on the edge set of K 6 . 3 / 15
Example Maker-Breaker triangle game on the edge set of K 6 . 3 / 15
Example Maker-Breaker triangle game on the edge set of K 6 . 3 / 15
Example Maker-Breaker triangle game on the edge set of K 6 . 3 / 15
Example Maker-Breaker triangle game on the edge set of K 6 . 3 / 15
Example Maker-Breaker triangle game on the edge set of K 6 . 3 / 15
Example Maker-Breaker triangle game on the edge set of K 6 . 3 / 15
Games on graphs ◮ Connectivity game: T – set of all spanning trees; ◮ Hamiltonicity game: H – set of all Hamiltonian cycles; ◮ H -game: G H – set of all copies of H , where H is a fixed graph (e.g., triangle game) 4 / 15
Games on graphs ◮ Connectivity game: T – set of all spanning trees; ◮ Hamiltonicity game: H – set of all Hamiltonian cycles; ◮ H -game: G H – set of all copies of H , where H is a fixed graph (e.g., triangle game) The games are played on the edge set of K n . What happens when n is large? 4 / 15
Games on graphs ◮ Connectivity game: T – set of all spanning trees; ◮ Hamiltonicity game: H – set of all Hamiltonian cycles; ◮ H -game: G H – set of all copies of H , where H is a fixed graph (e.g., triangle game) The games are played on the edge set of K n . What happens when n is large? All three games are easy Maker wins! 4 / 15
Games on graphs ◮ Connectivity game: T – set of all spanning trees; ◮ Hamiltonicity game: H – set of all Hamiltonian cycles; ◮ H -game: G H – set of all copies of H , where H is a fixed graph (e.g., triangle game) The games are played on the edge set of K n . What happens when n is large? All three games are easy Maker wins! To help Breaker, we can: 4 / 15
Games on graphs ◮ Connectivity game: T – set of all spanning trees; ◮ Hamiltonicity game: H – set of all Hamiltonian cycles; ◮ H -game: G H – set of all copies of H , where H is a fixed graph (e.g., triangle game) The games are played on the edge set of K n . What happens when n is large? All three games are easy Maker wins! To help Breaker, we can: ◮ Let Breaker claim more than one edge in each move – biased game , 4 / 15
Games on graphs ◮ Connectivity game: T – set of all spanning trees; ◮ Hamiltonicity game: H – set of all Hamiltonian cycles; ◮ H -game: G H – set of all copies of H , where H is a fixed graph (e.g., triangle game) The games are played on the edge set of K n . What happens when n is large? All three games are easy Maker wins! To help Breaker, we can: ◮ Let Breaker claim more than one edge in each move – biased game , ◮ Randomly remove some of the edges of the base graph before the game starts – random game . 4 / 15
Biased game Biased game (1 : b ) – Maker claims 1, and Breaker claims b edges per move. Introduced in [Chv´ atal-Erd˝ os 1978]. 5 / 15
Biased game Biased game (1 : b ) – Maker claims 1, and Breaker claims b edges per move. Introduced in [Chv´ atal-Erd˝ os 1978]. As b is increased, Breaker gains advantage... 5 / 15
Biased game Biased game (1 : b ) – Maker claims 1, and Breaker claims b edges per move. Introduced in [Chv´ atal-Erd˝ os 1978]. As b is increased, Breaker gains advantage... For a game F , the threshold bias b F is the largest integer such that Maker can win biased (1 : b F ) game. 5 / 15
Biased game Biased game (1 : b ) – Maker claims 1, and Breaker claims b edges per move. Introduced in [Chv´ atal-Erd˝ os 1978]. As b is increased, Breaker gains advantage... For a game F , the threshold bias b F is the largest integer such that Maker can win biased (1 : b F ) game. n ◮ Connectivity game: b T = (1 + o (1)) log n , [Gebauer-Szab´ o 2009], [Chv´ atal-Erd˝ os 1978] 5 / 15
Biased game Biased game (1 : b ) – Maker claims 1, and Breaker claims b edges per move. Introduced in [Chv´ atal-Erd˝ os 1978]. As b is increased, Breaker gains advantage... For a game F , the threshold bias b F is the largest integer such that Maker can win biased (1 : b F ) game. n ◮ Connectivity game: b T = (1 + o (1)) log n , [Gebauer-Szab´ o 2009], [Chv´ atal-Erd˝ os 1978] n ◮ Hamiltonicity game: b H = (1 + o (1)) log n , [Krivelevich 2011], [Chv´ atal-Erd˝ os 1978] 5 / 15
Biased game Biased game (1 : b ) – Maker claims 1, and Breaker claims b edges per move. Introduced in [Chv´ atal-Erd˝ os 1978]. As b is increased, Breaker gains advantage... For a game F , the threshold bias b F is the largest integer such that Maker can win biased (1 : b F ) game. n ◮ Connectivity game: b T = (1 + o (1)) log n , [Gebauer-Szab´ o 2009], [Chv´ atal-Erd˝ os 1978] n ◮ Hamiltonicity game: b H = (1 + o (1)) log n , [Krivelevich 2011], [Chv´ atal-Erd˝ os 1978] 1 � � ◮ H -game: b G H = Θ n m 2( H ) . [Bednarska-� Luczak, 2000] e ( H ′ ) − 1 ...where m 2 ( H ) = max H ′ ⊆ H , v ( H ′ ) ≥ 3 v ( H ′ ) − 2 . 5 / 15
Biased game The so-called random intuition [Erd˝ os] in positional games suggests that the outcome of the same positional game ◮ played by two smart players, and ◮ played by two “stupid” (random) players, could be the same. 6 / 15
Biased game The so-called random intuition [Erd˝ os] in positional games suggests that the outcome of the same positional game ◮ played by two smart players, and ◮ played by two “stupid” (random) players, could be the same. n Connectivity game: b T ∼ log n , so density of Maker’s edges at the end of the (1 : b T ) connectivity b T + 1 ∼ log n 1 game is n 6 / 15
Biased game The so-called random intuition [Erd˝ os] in positional games suggests that the outcome of the same positional game ◮ played by two smart players, and ◮ played by two “stupid” (random) players, could be the same. n Connectivity game: b T ∼ log n , so density of Maker’s edges at the end of the (1 : b T ) connectivity b T + 1 ∼ log n 1 game is = pr. threshold for connectivity in G ( n , p ). n 6 / 15
Biased game The so-called random intuition [Erd˝ os] in positional games suggests that the outcome of the same positional game ◮ played by two smart players, and ◮ played by two “stupid” (random) players, could be the same. n Connectivity game: b T ∼ log n , so density of Maker’s edges at the end of the (1 : b T ) connectivity b T + 1 ∼ log n 1 game is = pr. threshold for connectivity in G ( n , p ). n 1 m 2( H ) . Clique game: b G H ∼ n 1 But the threshold for appearance of H in G ( n , p ) is n − m ( H ) , e ( G ′ ) ...where m ( G ) = max G ′ ⊆ G v ( G ′ ) . 6 / 15
Random game 7 / 15
Random game To help Breaker, we randomly remove some of the edges of the base graph before the game starts – each edge is included with probability p , independently. 7 / 15
Random game To help Breaker, we randomly remove some of the edges of the base graph before the game starts – each edge is included with probability p , independently. So, the game is actually played on the edge set of a random graph G ( n , p ) . 7 / 15
Random game To help Breaker, we randomly remove some of the edges of the base graph before the game starts – each edge is included with probability p , independently. So, the game is actually played on the edge set of a random graph G ( n , p ) . If game F is Maker’s win when played with bias (1 : 1) on K n , the threshold probability p F is the probability at which an almost sure Breaker’s win turns into an almost sure Maker’s win. 7 / 15
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