It is well-known that the G(n, p) model of random graphs undergoes a dramatic change around p = 1 n . It is here that the random graph, almost surely, contains cycles, and here it first acquires a giant (i.e., order Ω(n)) connected component. Several years ago, Linial and Meshulam have introduced the Y d (n, p) model, a probability space of n-vertex d-dimensional simplicial complexes, where Y 1 (n, p) coincides with G(n, p). Within this model we prove a natural d-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real d-th homology of complexes from Y d (n, p). We also compute the real Betti numbers of Y d (n, p) for p = c/n. Finally, we establish the emergence of giant shadow at this threshold. (For d = 1 a giant shadow and a giant component are equivalent). Unlike the case for graphs, for d ≥ 2 the emergence of the giant shadow is a first order phase transition.

Abstract. The random 2-dimensional simplicial complex process starts with a complete graph on n vertices, and in every step a new 2-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to 1 as n → ∞, the first homology group over Z vanishes at the very moment when all the edges are covered by triangular faces.

For a graph G, let pi(G),i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call (p0(G),...,p3(G)) the 3‐local profile of G. We investigate the set scriptS3⊂double-struckR4 of all vectors (p0,...,p3) that are arbitrarily close to the 3‐local profiles of arbitrarily large graphs. We give a full description of the projection of S3 to the (p0,p3) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p0+p3≥14. We also give a full description of the triangle‐free case, i.e. the intersection of S3 with the hyperplane p3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

Let r, s ≥ 2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph Kn is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r = 2 or s = 2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.

This article surveys some of the work done in recent years on random simplicial complexes. We mostly consider higher-dimensional analogs of the well known phase transition in G(n, p) theory that occurs at p = 1 n . Our main objective is to provide a more streamlined and unified perspective of some of the papers in this area.