Thresholds and Expectation-Thresholds of Monotone Properties with Small Minterms

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ 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 $\mu_p$ on the power set of $N$, where $\mu_p(\cal{A}) := Pr[N_p \in \cal{A}]$ for $\cal{A} \subseteq 2^N$.In this paper we study monotone (upward-closed) families $\cal{A}$ for which all minimal sets in $cal{A}$ have size at most $k$, for some positive integer $k$. We prove that for such a family $\mu_p(\cal{A}) / p^k $ 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 $\cal{A}$ either a substantial subfamily $\cal{A}_0$ 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.

On linear programming duality and Landau’s characterization of tournament

H. G. Landau has characterized those integer-sequences S = (s1, s2, . . . , sn) which can arise as score-vectors in an ordinary round-robin tournament among n contestants [17]. If s1 ≤ s2 ≤ · · · ≤ sn, the relevant conditions are expressed simply by the inequalities: