We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nea... more We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nearly optimal p (up to a factor). In particular, we show that given t = (1 − o(1))np Hamilton cycles C1,…,Ct, each of which is oriented arbitrarily, a digraph ∼(n, p) w.h.p. contains edge disjoint copies of C1,…,Ct, provided . We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph ∼(n, p) w.h.p. contains (1 ± o(1))n!pn copies of C, provided .
In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$.... more In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$. We prove that for every $\delta>0$ and large enough $n$, there exists a constant $k$ for which if $\delta(G)\geq \delta n$ and $\chi(G)\geq k$, then Maker can build an odd cycle in the $(1:b)$ game for $b=O\left(\frac{n}{\log^2 n}\right)$. We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, Łuczak and Rödl: is it true that for any positive constants $t$ and $b$, there exists an integer $k$ such that for every graph $G$, if $\chi(G)\geq k$, then Maker can build a graph which is not $t$-colorable, in the $(1:b)$ Maker-Breaker game played on the vertices of $G$?
We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in... more We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for g...
Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-incr... more Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path.In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In parti...
We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2... more We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2007. A p-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is p). We analyze the random-turn version of several classical Maker-Breaker games such as the game Box (introduced by Chvátal and Erdős in 1987), the Hamilton cycle game and the k-vertex-connectivity game (both played on the edge set of K n). For each of these games we provide each of the players with a (randomized) efficient strategy which typically ensures his win in the asymptotic order of the minimum value of p for which he typically wins the game, assuming optimal strategies of both players.
We present a general approach connecting biased Maker-Breaker games and problems about local resi... more We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for b = o (√ n), Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a (1 : b) game on E(K n). As another application, we show that for b = Θ (n/ ln n), playing a (1 : b) game on E(K n), Maker can build a graph which contains copies of all spanning trees having maximum degree ∆ = O(1) with a bare path of linear length (a bare path in a tree T is a path with all interior vertices of degree exactly two in T).
A k-uniform hypergraph H contains a Hamilton ℓ-cycle, if there is a cyclic ordering of the vertic... more A k-uniform hypergraph H contains a Hamilton ℓ-cycle, if there is a cyclic ordering of the vertices of H such that the edges of the cycle are segments of length k in this ordering and any two consecutive edges f i , f i+1 share exactly ℓ vertices. We consider problems about packing and counting Hamilton ℓ-cycles in hypergraphs of large minimum degree. Given a hypergraph H, for a d-subset A ⊆ V (H), we denote by d H (A) the number of distinct edges f ∈ E(H) for which A ⊆ f , and set δ d (H) to be the minimum d H (A) over all A ⊆ V (H) of size d. We show that if a k-uniform hypergraph on n vertices H satisfies δ k−1 (H) ≥ αn for some α > 1/2, then for every ℓ < k/2 H contains (1 − o(1)) n • n! • α ℓ!(k−2ℓ)! n k−ℓ Hamilton ℓ-cycles. The exponent above is easily seen to be optimal. In addition, we show that if δ k−1 (H) ≥ αn for α > 1/2, then H contains f (α)n edge-disjoint Hamilton ℓ-cycles for an explicit function f (α) > 0. For the case where every (k − 1)-tuple X ⊂ V (H) satisfies d H (X) ∈ (α ± o(1))n, we show that H contains edge-disjoint Haimlton ℓ-cycles which cover all but o (|E(H)|) edges of H. As a tool we prove the following result which might be of independent interest: For a bipartite graph G with both parts of size n, with minimum degree at least δn, where δ > 1/2, and for p = ω(log n/n) the following holds. If G contains an r-factor for r = Θ(n), then by retaining edges of G with probability p independently at random, w.h.p the resulting graph contains a (1 − o(1))rp-factor.
A graph G is said to be H(n, ∆)-universal if it contains every graph on at most n vertices with m... more A graph G is said to be H(n, ∆)-universal if it contains every graph on at most n vertices with maximum degree at most ∆. It is known that for any ε > 0 and any natural number ∆ there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely H((1 − ε)n, ∆)universal for p ≥ c(log n/n) 1/∆. Bypassing this natural boundary, we show that for ∆ ≥ 3 the same conclusion holds when p ≫ n − 1 ∆−1 log 5 n.
The Seventh European Conference on Combinatorics, Graph Theory and Applications, 2013
For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the comple... more For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree, then Maker can win this game within n + 1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n − 1 moves and provide non-trivial examples of families of trees which Maker cannot build in n − 1 moves.
In this paper we analyze biased Maker-Breaker games and Avoider-Enforcer games, both played on th... more In this paper we analyze biased Maker-Breaker games and Avoider-Enforcer games, both played on the edge set of a random board G ∼ G(n, p). In Maker-Breaker games there are two players, denoted by Maker and Breaker. In each round, Maker claims one previously unclaimed edge of G and Breaker responds by claiming b previously unclaimed edges. We consider the Hamiltonicity game, the perfect matching game and the k-vertexconnectivity game, where Maker's goal is to build a graph which possesses the relevant property. Avoider-Enforcer games are the reverse analogue of Maker-Breaker games with a slight modification, where the two players claim at least 1 and at least b previously unclaimed edges per move, respectively, and Avoider aims to avoid building a graph which possesses the relevant property. Maker-Breaker games are known to be "bias-monotone", that is, if Maker wins the (1, b) game, he also wins the (1, b − 1) game. Therefore, it makes sense to define the critical bias of a game, b * , to be the "breaking point" of the game. That is, Maker wins the (1, b) game whenever b ≤ b * and loses otherwise. An analogous definition of the critical bias exists for Avoider-Enforcer games: here, the critical bias of a game b * is such that Avoider wins the (1, b) game for every b > b * , and loses otherwise. We prove that, for every p = ω(ln n n), G ∼ G(n, p) is typically such that the critical bias for all the aforementioned Maker-Breaker games is asymptotically b * = np ln n. We also prove that in the case p = Θ(ln n n), the critical bias is b * = Θ(np ln n). These results settle a conjecture of Stojaković and Szabó. For Avoider-Enforcer games, we prove that for p = Ω(ln n n), the critical bias for all the aforementioned games is b * = Θ(np ln n).
We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider ... more We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property P. We focus on three natural properties for Maker's graph, namely being k-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k-vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojaković and Szabó.
Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and ... more Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k-uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.
Where a licence is displayed above, please note the terms and conditions of the licence govern yo... more Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.
We analyze Avoider-Enforcer games played on edge disjoint hypergraphs, providing an analog of the... more We analyze Avoider-Enforcer games played on edge disjoint hypergraphs, providing an analog of the classic and well known game Box, due to Chvátal and Erdős. We consider both strict and monotone versions of Avoider-Enforcer games, and for each version we give a sufficient condition to win for each player. We also present applications of our results to several general Avoider-Enforcer games.
We answer questions concerning an existence of almost precipitous ideals raised in [5]. It is sho... more We answer questions concerning an existence of almost precipitous ideals raised in [5]. It is shown that every successor of a regular cardinal can carry an almost precipitous ideal in a generic extension of L. In L[µ] every regular cardinal which is less than the measurable carries an almost precipitous non-precipitous ideal. Also, results of [4] are generalized-thus assumptions on precipitousness are replaced by those on ∞-semi precipitousness. 1 On semi precipitous and almost precipitous ideals Definition 1.1 Let κ be a regular uncountable cardinal, τ a ordinal and I a κ-complete ideal over κ. We call I τ-almost precipitous iff every generic ultrapower of I is wellfounded up to the image of τ. Clearly, any such I is τ-almost precipitous for each τ < κ. Also, note if τ ≥ (2 κ) + and I is τ-almost precipitous, then I is precipitous. Definition 1.2 Let κ be a regular uncountable cardinal. We call κ almost precipitous iff for each τ < (2 κ) + there is τ-almost precipitous ideal over κ.
Given a tree T = (V, E) on n vertices, we consider the (1 : q) Maker-Breaker tree embedding game ... more Given a tree T = (V, E) on n vertices, we consider the (1 : q) Maker-Breaker tree embedding game T n. The board of this game is the edge set of the complete graph on n vertices. Maker wins T n if and only if he is able to claim all edges of a copy of T. We prove that there exist real numbers α, ε > 0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most n ε , Maker has a winning strategy for the (1 : q) game T n , for every q ≤ n α. Moreover, we prove that Maker can win this game within n + o(n) moves which is clearly asymptotically optimal.
We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nea... more We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nearly optimal p (up to a factor). In particular, we show that given t = (1 − o(1))np Hamilton cycles C1,…,Ct, each of which is oriented arbitrarily, a digraph ∼(n, p) w.h.p. contains edge disjoint copies of C1,…,Ct, provided . We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph ∼(n, p) w.h.p. contains (1 ± o(1))n!pn copies of C, provided .
In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$.... more In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$. We prove that for every $\delta>0$ and large enough $n$, there exists a constant $k$ for which if $\delta(G)\geq \delta n$ and $\chi(G)\geq k$, then Maker can build an odd cycle in the $(1:b)$ game for $b=O\left(\frac{n}{\log^2 n}\right)$. We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, Łuczak and Rödl: is it true that for any positive constants $t$ and $b$, there exists an integer $k$ such that for every graph $G$, if $\chi(G)\geq k$, then Maker can build a graph which is not $t$-colorable, in the $(1:b)$ Maker-Breaker game played on the vertices of $G$?
We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in... more We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for g...
Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-incr... more Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path.In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In parti...
We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2... more We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2007. A p-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is p). We analyze the random-turn version of several classical Maker-Breaker games such as the game Box (introduced by Chvátal and Erdős in 1987), the Hamilton cycle game and the k-vertex-connectivity game (both played on the edge set of K n). For each of these games we provide each of the players with a (randomized) efficient strategy which typically ensures his win in the asymptotic order of the minimum value of p for which he typically wins the game, assuming optimal strategies of both players.
We present a general approach connecting biased Maker-Breaker games and problems about local resi... more We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for b = o (√ n), Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a (1 : b) game on E(K n). As another application, we show that for b = Θ (n/ ln n), playing a (1 : b) game on E(K n), Maker can build a graph which contains copies of all spanning trees having maximum degree ∆ = O(1) with a bare path of linear length (a bare path in a tree T is a path with all interior vertices of degree exactly two in T).
A k-uniform hypergraph H contains a Hamilton ℓ-cycle, if there is a cyclic ordering of the vertic... more A k-uniform hypergraph H contains a Hamilton ℓ-cycle, if there is a cyclic ordering of the vertices of H such that the edges of the cycle are segments of length k in this ordering and any two consecutive edges f i , f i+1 share exactly ℓ vertices. We consider problems about packing and counting Hamilton ℓ-cycles in hypergraphs of large minimum degree. Given a hypergraph H, for a d-subset A ⊆ V (H), we denote by d H (A) the number of distinct edges f ∈ E(H) for which A ⊆ f , and set δ d (H) to be the minimum d H (A) over all A ⊆ V (H) of size d. We show that if a k-uniform hypergraph on n vertices H satisfies δ k−1 (H) ≥ αn for some α > 1/2, then for every ℓ < k/2 H contains (1 − o(1)) n • n! • α ℓ!(k−2ℓ)! n k−ℓ Hamilton ℓ-cycles. The exponent above is easily seen to be optimal. In addition, we show that if δ k−1 (H) ≥ αn for α > 1/2, then H contains f (α)n edge-disjoint Hamilton ℓ-cycles for an explicit function f (α) > 0. For the case where every (k − 1)-tuple X ⊂ V (H) satisfies d H (X) ∈ (α ± o(1))n, we show that H contains edge-disjoint Haimlton ℓ-cycles which cover all but o (|E(H)|) edges of H. As a tool we prove the following result which might be of independent interest: For a bipartite graph G with both parts of size n, with minimum degree at least δn, where δ > 1/2, and for p = ω(log n/n) the following holds. If G contains an r-factor for r = Θ(n), then by retaining edges of G with probability p independently at random, w.h.p the resulting graph contains a (1 − o(1))rp-factor.
A graph G is said to be H(n, ∆)-universal if it contains every graph on at most n vertices with m... more A graph G is said to be H(n, ∆)-universal if it contains every graph on at most n vertices with maximum degree at most ∆. It is known that for any ε > 0 and any natural number ∆ there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely H((1 − ε)n, ∆)universal for p ≥ c(log n/n) 1/∆. Bypassing this natural boundary, we show that for ∆ ≥ 3 the same conclusion holds when p ≫ n − 1 ∆−1 log 5 n.
The Seventh European Conference on Combinatorics, Graph Theory and Applications, 2013
For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the comple... more For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree, then Maker can win this game within n + 1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n − 1 moves and provide non-trivial examples of families of trees which Maker cannot build in n − 1 moves.
In this paper we analyze biased Maker-Breaker games and Avoider-Enforcer games, both played on th... more In this paper we analyze biased Maker-Breaker games and Avoider-Enforcer games, both played on the edge set of a random board G ∼ G(n, p). In Maker-Breaker games there are two players, denoted by Maker and Breaker. In each round, Maker claims one previously unclaimed edge of G and Breaker responds by claiming b previously unclaimed edges. We consider the Hamiltonicity game, the perfect matching game and the k-vertexconnectivity game, where Maker's goal is to build a graph which possesses the relevant property. Avoider-Enforcer games are the reverse analogue of Maker-Breaker games with a slight modification, where the two players claim at least 1 and at least b previously unclaimed edges per move, respectively, and Avoider aims to avoid building a graph which possesses the relevant property. Maker-Breaker games are known to be "bias-monotone", that is, if Maker wins the (1, b) game, he also wins the (1, b − 1) game. Therefore, it makes sense to define the critical bias of a game, b * , to be the "breaking point" of the game. That is, Maker wins the (1, b) game whenever b ≤ b * and loses otherwise. An analogous definition of the critical bias exists for Avoider-Enforcer games: here, the critical bias of a game b * is such that Avoider wins the (1, b) game for every b > b * , and loses otherwise. We prove that, for every p = ω(ln n n), G ∼ G(n, p) is typically such that the critical bias for all the aforementioned Maker-Breaker games is asymptotically b * = np ln n. We also prove that in the case p = Θ(ln n n), the critical bias is b * = Θ(np ln n). These results settle a conjecture of Stojaković and Szabó. For Avoider-Enforcer games, we prove that for p = Ω(ln n n), the critical bias for all the aforementioned games is b * = Θ(np ln n).
We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider ... more We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property P. We focus on three natural properties for Maker's graph, namely being k-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k-vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojaković and Szabó.
Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and ... more Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k-uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.
Where a licence is displayed above, please note the terms and conditions of the licence govern yo... more Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.
We analyze Avoider-Enforcer games played on edge disjoint hypergraphs, providing an analog of the... more We analyze Avoider-Enforcer games played on edge disjoint hypergraphs, providing an analog of the classic and well known game Box, due to Chvátal and Erdős. We consider both strict and monotone versions of Avoider-Enforcer games, and for each version we give a sufficient condition to win for each player. We also present applications of our results to several general Avoider-Enforcer games.
We answer questions concerning an existence of almost precipitous ideals raised in [5]. It is sho... more We answer questions concerning an existence of almost precipitous ideals raised in [5]. It is shown that every successor of a regular cardinal can carry an almost precipitous ideal in a generic extension of L. In L[µ] every regular cardinal which is less than the measurable carries an almost precipitous non-precipitous ideal. Also, results of [4] are generalized-thus assumptions on precipitousness are replaced by those on ∞-semi precipitousness. 1 On semi precipitous and almost precipitous ideals Definition 1.1 Let κ be a regular uncountable cardinal, τ a ordinal and I a κ-complete ideal over κ. We call I τ-almost precipitous iff every generic ultrapower of I is wellfounded up to the image of τ. Clearly, any such I is τ-almost precipitous for each τ < κ. Also, note if τ ≥ (2 κ) + and I is τ-almost precipitous, then I is precipitous. Definition 1.2 Let κ be a regular uncountable cardinal. We call κ almost precipitous iff for each τ < (2 κ) + there is τ-almost precipitous ideal over κ.
Given a tree T = (V, E) on n vertices, we consider the (1 : q) Maker-Breaker tree embedding game ... more Given a tree T = (V, E) on n vertices, we consider the (1 : q) Maker-Breaker tree embedding game T n. The board of this game is the edge set of the complete graph on n vertices. Maker wins T n if and only if he is able to claim all edges of a copy of T. We prove that there exist real numbers α, ε > 0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most n ε , Maker has a winning strategy for the (1 : q) game T n , for every q ≤ n α. Moreover, we prove that Maker can win this game within n + o(n) moves which is clearly asymptotically optimal.
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