Spaces of completions of elementary theories and convergence laws for random hypergraphs

Abstract

Consider the binomial model G d+1 (n, p) of the random (d + 1)uniform hypergraph on n vertices, where each edge is present, independently of one another, with probability p : N → [0, 1]. We prove that, for all logarithmo-exponential p n −d+ , the probabilities of all elementary properties of hypergraphs converge, with particular emphasis in the ranges p(n) ∼ C/n d and p(n) ∼ C log(n)/n d . The exposition is unified by constructing, for each such function p, the topological space of all completions of its almost sure theory. This space turns out to be compact, metrizable and totally disconnected, but further properties depend on the range of p. The convergence of the probabilities of elementary properties is associated with a borelian probability measure on the space.

Limiting probabilities of first order properties of random sparse graphs and hypergraphs

“…The following is an analog of Lynch's convergence law for random hypergraphs and can be found in [11,Proposition 6.4] and in more detail for more general relational structures in [8].…”