Thresholds and Expectation-Thresholds of Monotone Properties with Small Minterms
نویسندگان
چکیده
Let N be a finite set, let p ∈ (0, 1), and let Np denote a random binomial subset of N where every element of N is taken to belong to the subset independently with probability p. This defines a product measure μp on the power set of N , where μp(A) := Pr[Np ∈ A] for A ⊆ 2N . In this paper we study monotone (upward-closed) families A for which all minimal sets in A have size at most k, for some positive integer k. We prove that for such a family μp(A)/p is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables one to identify in A either a substantial subfamily A0 for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than k. Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using linear programming duality. This is related to the threshold conjecture of Kahn and Kalai. ∗Research supported in part by I.S.F. grant 0398246, BSF grant 2010247 and MINERVA grant 712023. †Research supported in part by BSF grant 2010247. the electronic journal of combinatorics 22(2) (2015), #P2.39 1
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عنوان ژورنال:
- Electr. J. Comb.
دوره 22 شماره
صفحات -
تاریخ انتشار 2015