B. Kreuter, Small sublattices in random subsets of Boolean lattices, Random Struct. Alg., 13, 383407 (1998). Home/Search Document Not in Database Summary Related Articles Check |
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Maximum Antichains in Random Subsets of a Finite Set - Osthus (Correct)
....the isolated vertices in V 2 is thus a.s. an antichain whose size satis es the requirements. 2 The remainder of this note is organized as follows: in Section 2 we derive properties of regular chains in P(n; p) which will enable us to prove Theorem 2 in Section 3. We refer the reader to Kreuter [5] for further results on the evolution of P(n; p) and to Brightwell [2] for a survey on other models of random posets. 2 The chain cover To prove the upper bound in Theorem 2 we need a covering of P(n; p) with chains which is well behaved in the sense that for most elements a 2 P(n; p) the ....
B. Kreuter, Small sublattices in random subsets of Boolean lattices, Random Struct. Alg. 13, 383407 (1998).
The Length of Random Subsets of Boolean Lattices - Kohayakawa, al. (1998) Self-citation (Kreuter) (Correct)
....are pairwise comparable and the length of a chain is its cardinality minus one. Rnyi was the rst to investigate the behaviour of this random variable: answering a question of Erds, in [12] he determined the threshold for the existence of a chain of length one in P(n; p) Recently, Kreuter [10] obtained (amongst other results) thresholds for the appearance of chains of any xed length. This raises the question how the length evolves if we increase p beyond the range considered in [10] In this paper we answer this question by studying the behaviour of certain regular chains in P(n; ....
....he determined the threshold for the existence of a chain of length one in P(n; p) Recently, Kreuter [10] obtained (amongst other results) thresholds for the appearance of chains of any xed length. This raises the question how the length evolves if we increase p beyond the range considered in [10]. In this paper we answer this question by studying the behaviour of certain regular chains in P(n; p) Another important parameter of P(n; p) is the cardinality of its maximum antichain, also called the width. This is studied in [9, 11] where the results proven in [11] are based on the ....
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B. Kreuter, Small sublattices in random subsets of Boolean lattices, Random Struct. Alg., 13, 383407 (1998).
The Length of Random Subsets of Boolean Lattices - Kohayakawa, Kreuter, Osthus (1998) Self-citation (Kreuter) (Correct)
.... of P(n; p) for which we write length P(n; p) is de ned to be the length of the maximum chain contained in P(n; p) Rnyi was the rst to investigate the behaviour of this random variable: in [12] he determined the threshold for the existence of a chain of length one in P(n; p) Recently, Kreuter [10] obtained (amongst other results) thresholds for the appearance of chains of any xed length. Thus in this paper we con ne our attention to the evolution of the length of P(n; p) when Research supported by PROBRAL project number 026 95, a CAPES DAAD exchange programme. y Partially supported ....
....tending to one as n tends to in nity . This paper is organised as follows. In the next section, we state our main results, Theorems 1 and 4. Our main technical tool is given in Section 3, where we state and prove Lemma 7. Theorems 1 and 4 are proved in Sections 4 and 5. 2 Results Kreuter [10] proved that the length of P(n; p) is constant when p decreases exponentially. The following theorem tells us about the length of P(n; p) when p is superexponential: Theorem 1 Let k = k(n) n 1 be an integer valued function tending to in nity. Then lim n 1 P( length P(n; p) k 1 ) 0 if ....
B. Kreuter, Small sublattices in random subsets of Boolean lattices, Random Struct. Alg., to appear. 18
The Width of Random Subsets of Boolean Lattices - Kohayakawa, Kreuter (1998) (1 citation) Self-citation (Kreuter) (Correct)
....jP(n; p)j i 1 O i 2 jj r 8 : The proof of Theorem 1 is given in the next section. For results concerning the length of P(n; p) i.e. the cardinality of a longest chain, the reader is referred to [1] The threshold function for small sublattices in P(n; p) are investigated in [2], which may be thought of as a modern sequel to R enyi s pioneering work [3] 2 Proofs 2.1 Proof of Theorem 1 Before giving the proof of Theorem 1, we state some asymptotics involving binomial coefficients. To simplify the notation, we shall often omit the b c and d e signs whenever they are not ....
B. Kreuter. Small sublattices in random subsets of Boolean lattices. Submitted, 1997.
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