We prove the existence conjecture for combinatorial designs, answering a question of
Steiner from 1853. More generally, we show that the natural divisibility conditions are …

Topology of random simplicial complexes: a survey Page 220 Contemporary Mathematics
Volume 620 , 2014 http://dx. doi. org/10.1090/conm/620/12367 Topology of random simplicial …

In this paper we propose a model of random simplicial complexes with randomness in all
dimensions. We start with a set of n vertices and retain each of them with probability p 0; on …

For every k≥ 1, the k-th cohomology group Hk (X, ℚ) of the random flag complex X∼ X (n, p)
passes through two phase transitions: one where it appears and one where it vanishes. We …

Let Δ _ n-1 denote the (n-1)-dimensional simplex. Let Y be a random d-dimensional
subcomplex of Δ _ n-1 obtained by starting with the full (d-1)-dimensional skeleton of Δ _ n-1 …

We review a collection of models of random simplicial complexes together with some of the
most exciting phenomena related to them. We do not attempt to cover all existing models …

We consider higher-dimensional generalizations of the normalized Laplacian and the
adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam model X k …

We compute the homology of random Čech complexes over a homogeneous Poisson
process on the d‐dimensional torus, and show that there are, coarsely, two phase …

We describe a higher-dimensional generalization of edge expansion from graphs to
simplicial complexes, which we discuss as a type of co-isoperimetric inequality. The main …

Abstract We study the Linial–Meshulam model of random two-dimensional simplicial
complexes. One of our main results states that for p≪ n− 1 a random 2-complex Y collapses …