D ERANDOMIZATION IS OFTEN USELESS If we apply Lemma for the family T of all spanning trees it gives ( since n n − 2 ( q + 1 ) − ( n − 1 ) < 1 / 2 ) . q T ( n ) < n / 2 , The real threshold is q T ( n ) = ( 1 + o ( 1 )) n / log n and it agrees with the random graph heuristic! Here the problem follows from the fact that the connectivity threshold for random graphs cannot be estimated by bounding the first moment of X T ( n , q ) .
Y ET ANOTHER NICE IDEA Go greedy!
Y ET ANOTHER NICE IDEA Go greedy! Idea: � n If only a very few graphs on n vertices and M = ( q + 1 ) �� 2 edges do not have property A , then Maker can win MB ( n , q , A ) playing greedily, i.e. he should always choose the edge which maximizes the chance that his final graph will have property A .
Y ET ANOTHER NICE IDEA Go greedy! Idea: � n If only a very few graphs on n vertices and M = ( q + 1 ) �� 2 edges do not have property A , then Maker can win MB ( n , q , A ) playing greedily, i.e. he should always choose the edge which maximizes the chance that his final graph will have property A . In a similar way, Waiter can win WC ( n , q , A ) proposing q + 1 edges which favor property A .
Y ET ANOTHER NICE IDEA Go greedy! Idea: � n If only a very few graphs on n vertices and M = ( q + 1 ) �� 2 edges do not have property A , then Maker can win MB ( n , q , A ) playing greedily, i.e. he should always choose the edge which maximizes the chance that his final graph will have property A . In a similar way, Waiter can win WC ( n , q , A ) proposing q + 1 edges which favor property A . The main problem here: in order this strategy to work the probability that G ( n , M ) has no property A must be very, very small, i.e. smaller than exp ( − c 0 M ) for some absolute constant c 0 .
G REEDY L EMMA B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK For every c > 0 there exists (an explicit) 0 < b < 1 such that Maker [Waiter] has a strategy that in the first � n b / ( q + 1 ) steps of the game Maker’s [Client’s] graph � 2 has property A , provided only that Pr [ G ( n , M ) has not A ] ≤ exp ( − cM ) , � n for M = b / ( q + 1 ) . � 2
G ( n , M ) REVISITED Let H be a given (small) subgraph and let e ( F ) − 1 m 2 ( H ) = max v ( F ) − 2 . F ⊆ H
G ( n , M ) REVISITED Let H be a given (small) subgraph and let e ( F ) − 1 m 2 ( H ) = max v ( F ) − 2 . F ⊆ H If m 2 ( H ) = e ( H ) − 1 v ( H ) − 2 then H is 2-balanced.
G ( n , M ) REVISITED Let H be a given (small) subgraph and let e ( F ) − 1 m 2 ( H ) = max v ( F ) − 2 . F ⊆ H If m 2 ( H ) = e ( H ) − 1 v ( H ) − 2 then H is 2-balanced. Example: K 3 is 2-balanced with m 2 ( K 3 ) = 2.
G ( n , M ) REVISITED T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI ’90 For every H and a constant a > 0 there exists a constant c > 0 such that Pr ( G ( n , M ) �⊇ H ) ≤ exp ( − cM ) , provided M ≥ an 2 − m 2 ( H ) . T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI ’90 For every 2-balanced H and a constant a > 0 there exist constants b > 0 and c > 0 such that the probability that G ( n , M ) contains fewer than bM edge disjoint copies of H is less than exp ( − cM ) , provided M ≥ an 2 − m 2 ( H ) .
G ( n , M ) REVISITED T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI ’90 For every H and a constant a > 0 there exists a constant c > 0 such that Pr ( G ( n , M ) �⊇ H ) ≤ exp ( − cM ) , provided M ≥ an 2 − m 2 ( H ) . T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI ’90 For every 2-balanced H and a constant a > 0 there exist constants b > 0 and c > 0 such that the probability that G ( n , M ) contains fewer than bM edge disjoint copies of H is less than exp ( − cM ) , provided M ≥ an 2 − m 2 ( H ) .
G ( n , M ) REVISITED T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI For every 2-balanced H and a constant a > 0 there exist constants b > 0 and c > 0 such that the probability that G ( n , M ) contains fewer than bM edge disjoint copies of H is less than exp ( − cM ) , provided M ≥ an 2 − m 2 ( H ) .
G ( n , M ) REVISITED T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI For every 2-balanced H and a constant a > 0 there exist constants b > 0 and c > 0 such that the probability that G ( n , M ) contains fewer than bM edge disjoint copies of H is less than exp ( − cM ) , provided M ≥ an 2 − m 2 ( H ) . Remark 1: If we want the expected number of H to be of the order M we need to have M ≥ an 2 − m 2 ( H ) .
G ( n , M ) REVISITED T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI For every 2-balanced H and a constant a > 0 there exist constants b > 0 and c > 0 such that the probability that G ( n , M ) contains fewer than bM edge disjoint copies of H is less than exp ( − cM ) , provided M ≥ an 2 − m 2 ( H ) . Remark 1: If we want the expected number of H to be of the order M we need to have M ≥ an 2 − m 2 ( H ) . Remark 2: Typically, c cannot be made arbitrarily large, even for large a . For instance, the probability that G ( n , M ) �⊇ K 3 is larger than ( 1 / 2 + o ( 1 )) M , whenever M = o ( n 2 ) .
T RIANGLES T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI ’89 If M ≥ 10 n 3 / 2 then with probability at least exp ( − M / 5 ) G ( n , M ) contains at least M / 100 edge disjoint triangles. L EMMA B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK ’14+ � n / ( q + 1 ) steps his � Maker has a strategy that in the first 0 . 01 2 graph has property A , provided Pr [ G ( n , M ) has not A ] ≤ exp ( − M / 5 ) , � n for M = 0 . 01 / ( q + 1 ) . � 2 C OROLLARY / ( q + 1 ) ≥ 1000 n 3 / 2 , i.e. when q ≤ 0 . 00001 √ n , then � n If M = � 2 Maker can win MB ( n , q , K 3 ) .
T RIANGLES T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI ’89 If M ≥ 10 n 3 / 2 then with probability at least exp ( − M / 5 ) G ( n , M ) contains at least M / 100 edge disjoint triangles. L EMMA B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK ’14+ � n / ( q + 1 ) steps his � Maker has a strategy that in the first 0 . 01 2 graph has property A , provided Pr [ G ( n , M ) has not A ] ≤ exp ( − M / 5 ) , � n for M = 0 . 01 / ( q + 1 ) . � 2 C OROLLARY / ( q + 1 ) ≥ 1000 n 3 / 2 , i.e. when q ≤ 0 . 00001 √ n , then � n If M = � 2 Maker can win MB ( n , q , K 3 ) .
T RIANGLES T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI ’89 If M ≥ 10 n 3 / 2 then with probability at least exp ( − M / 5 ) G ( n , M ) contains at least M / 100 edge disjoint triangles. L EMMA B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK ’14+ � n / ( q + 1 ) steps his � Maker has a strategy that in the first 0 . 01 2 graph has property A , provided Pr [ G ( n , M ) has not A ] ≤ exp ( − M / 5 ) , � n for M = 0 . 01 / ( q + 1 ) . � 2 C OROLLARY / ( q + 1 ) ≥ 1000 n 3 / 2 , i.e. when q ≤ 0 . 00001 √ n , then � n If M = � 2 Maker can win MB ( n , q , K 3 ) .
T RIANGLES T HEOREM J ANSON , Ł UCZAK , R UCI ´ NSKI ’89 If M ≥ 10 n 3 / 2 then with probability at least exp ( − M / 5 ) G ( n , M ) contains at least M / 100 edge disjoint triangles. L EMMA B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK ’14+ � n / ( q + 1 ) steps his � Maker has a strategy that in the first 0 . 01 2 graph has property A , provided Pr [ G ( n , M ) has not A ] ≤ exp ( − M / 5 ) , � n for M = 0 . 01 / ( q + 1 ) . � 2 In fact, Maker can win MB ( n , q , K 3 ) quite early, when fewer than 1% of all pairs have been used, and it can create at least 0 . 1 n 2 / 3 triangles.
A GOOD NEWS For every Maker-Breaker game MB ( n , q , H ) , when Maker tries to build a copy of graph H , this method gives the right (lower) bound for the threshold bias! T HEOREM B EDNARSKA , Ł UCZAK ’00 For each graph H there are constants C > c > 0 such that: ◮ if q < cn m 2 ( H ) , then Maker wins MB ( n , q , H ) ; ◮ if q > Cn m 2 ( H ) , then Breaker wins MB ( n , q , H ) .
A GOOD NEWS For every Maker-Breaker game MB ( n , q , H ) , when Maker tries to build a copy of graph H , this method gives the right (lower) bound for the threshold bias! T HEOREM B EDNARSKA , Ł UCZAK ’00 For each graph H there are constants C > c > 0 such that: ◮ if q < cn m 2 ( H ) , then Maker wins MB ( n , q , H ) ; ◮ if q > Cn m 2 ( H ) , then Breaker wins MB ( n , q , H ) . Remark: Note that for q < cn m 2 ( H ) Maker can quickly (!) �� n / ( q + 1 ) copies of H . � � create Ω 2
A BAD NEWS For Waiter-Client game MB ( n , q , H ) , this method gives the same lower bound for the threshold bias, but typically (in fact almost always) it is far off the right value. Example: In the case of H = K 3 , we have n ≫ √ n . C LAIM �� n For q < O ( n m 2 ( H ) ) Waiter can quickly create Ω / ( q + 1 ) � � 2 copies of H .
A BAD NEWS For Waiter-Client game MB ( n , q , H ) , this method gives the same lower bound for the threshold bias, but typically (in fact almost always) it is far off the right value. Example: In the case of H = K 3 , we have n ≫ √ n . However this method gives the following statement. C LAIM �� n For q < O ( n m 2 ( H ) ) Waiter can quickly create Ω / ( q + 1 ) � � 2 copies of H .
W HAT WE KNOW SO FAR We know that greedy method gives lower bounds for the threshold bias for both MB ( n , q , H ) and WC ( n , q , H ) which is of the correct order for the former one but does poorly for the latter one.
W HAT WE KNOW SO FAR We know that greedy method gives lower bounds for the threshold bias for both MB ( n , q , H ) and WC ( n , q , H ) which is of the correct order for the former one but does poorly for the latter one. The derandomization method provides upper bounds for the threshold bias for both MB ( n , q , H ) and WC ( n , q , H ) which typically, is very much off the former one but does quite well for the latter one (at least for H = K 3 ).
W HAT WE ARE ABOUT TO DO NEXT C ONJECTURE B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK For every q and H , in Waiter-Client q -biased game Waiter can always force at least � n �� F ⊆ H E X H n , ( q + 1 ) � � �� Ω min 2 copies of H . T HEOREM B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK The above conjecture is true for many ‘nice’ graphs H (such as, for instance, complete graphs K k ).
W HAT WE ARE ABOUT TO DO NEXT C ONJECTURE B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK For every q and H , in Waiter-Client q -biased game Waiter can always force at least � n �� F ⊆ H E X H n , ( q + 1 ) � � �� Ω min 2 copies of H . T HEOREM B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK The above conjecture is true for many ‘nice’ graphs H (such as, for instance, complete graphs K k ).
A USEFUL OBSERVATION C ONJECTURE B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK For every q and H , in Waiter-Client q -biased game Waiter can always force at least Ω( f ( n )) copies of H , where � n �� f ( n ) = min F ⊆ H E X H n , ( q + 1 ) � � . 2 C LAIM B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK By the greedy method, the above conjecture is true, when q is small enough. In fact, when the expected number f ( n ) is of the � n order of M ( n ) = / ( q + 1 ) , then Waiter can quickly force at � 2 least Ω( n ) copies of H in Client’s graph.
E XPECTATION OF X H Clearly, � n �� ( q + 1 )) ≍ n v ( H ) ( q + 1 ) − e ( H ) , E X H ( n , 2 where v ( H ) and e ( H ) denote the number of vertices and edges in H respectively, while ‘ ≍ ’ means up to constant factor which may depend on H .
T HE EXPECTATION OF X H E X H ≍ n v ( H ) ( q + 1 ) − e ( H ) .
T HE EXPECTATION OF X H E X H ≍ n v ( H ) ( q + 1 ) − e ( H ) . Let H ( e 1 , . . . , e k ) denote the graph obtained from H by removing k of its edges: e 1 , . . . , e k . Then, E X H ( e 1 ,..., e k ) ≍ E X H ( q + 1 ) k .
P ROOF OF M AIN R ESULT T HEOREM B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK If H is ‘nice’ Waiter can force in Client’s graph roughly as many � n copies of H as E X H ( n , / ( q + 1 )) . � 2
P ROOF OF M AIN R ESULT T HEOREM B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK If H is ‘nice’ Waiter can force in Client’s graph roughly as many � n copies of H as E X H ( n , / ( q + 1 )) . � 2 Proof We know (by the greedy method) that the above is true for 2-balanced H such that E X H is at least as large as � n / ( q + 1 ) . Thus, suppose that it is not the case. � 2
P ROOF OF M AIN R ESULT T HEOREM B EDNARSKA -B ZDE ¸ GA , H EFETZ , Ł UCZAK If H is ‘nice’ Waiter can force in Client’s graph roughly as many � n copies of H as E X H ( n , / ( q + 1 )) . � 2 Proof We know (by the greedy method) that the above is true for 2-balanced H such that E X H is at least as large as � n / ( q + 1 ) . Thus, suppose that it is not the case. � 2 Delete from H some of its edges such that H ( e 1 , . . . , e k ) is � n 2-balanced and E X H ( e 1 ,..., e k ) is at least as large as / ( q + 1 ) � 2 (one can do if H is ‘nice’ enough).
P ROOF OF M AIN R ESULT C LAIM If H is ‘nice’ enough then, for some α > 0, Waiter can quickly force in Client’s graph a family H ( e 1 , . . . , e k ) of α ′ E X H ( e 1 ,..., e k ) ≥ α E X H ( q + 1 ) k copies of H ( e 1 , . . . , e k ) such that: ( A ) Each of the missing pairs e 1 , . . . , e k belongs to just one copy of H ( e 1 , . . . , e k ) from H ( e 1 , . . . , e k ) ; ( B ) None of the missing pairs e 1 , . . . , e k has been offered by Waiter yet.
P ROOF OF M AIN R ESULT |H ( e 1 , . . . , e k − 1 , e k ) | ≥ α E X H ( q + 1 ) k
P ROOF OF M AIN R ESULT |H ( e 1 , . . . , e k − 1 , e k ) | ≥ α E X H ( q + 1 ) k |H ( e 1 , . . . , e k − 1 ) | ≥ α E X H ( q + 1 ) k − 1
P ROOF OF M AIN R ESULT |H ( e 1 , . . . , e k − 1 , e k ) | ≥ α E X H ( q + 1 ) k |H ( e 1 , . . . , e k − 1 ) | ≥ α E X H ( q + 1 ) k − 1 . . . |H| ≥ α E X H
P ROBLEM WITH THIS APPROACH Even ‘good-looking’ graphs may not be ‘nice’.
P ROBLEM WITH THIS APPROACH Even ‘good-looking’ graphs may not be ‘nice’. Example: H = K 5 − e .
P ROBLEM WITH THIS APPROACH Even ‘good-looking’ graphs may not be ‘nice’. Example: H = K 5 − e . E X K 5 − e = n 5 ( q + 1 ) − 9 ≍ 1 . � n Thus, q ≍ n 5 / 9 and M = / ( q + 1 ) ≍ n 13 / 9 . � 2 Consequently, we have to remove from K 5 − e three edges so that the expectation of the resulting graph jumps to n 15 / 9 ≫ n 13 / 9 .
K 5 − e Thus, H ( e 1 , e 2 , e 3 ) is C 5 ∪ e . But C 5 ∪ e is not 2-balanced!
K 5 − e Thus, H ( e 1 , e 2 , e 3 ) is C 5 ∪ e . But C 5 ∪ e is not 2-balanced! Indeed, m 2 ( C 5 ∪ e ) = e ( C 5 ∪ e ) − 1 v ( C 5 ∪ e ) − 2 = 4 m 2 ( C 3 ) = 2 > 4 but 3 . 3
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