Yuval Peled

On the 3-Local Profiles of Graphs

For a graph G, let piG,i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call p0G,...,p3G the 3-local profileof G. We investigate the set S3i¾?R4 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 Goodmans inequality p0+p3i¾?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 Razborovs flag algebra theory.

Integral Homology of Random Simplicial Complexes

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

A Random Triadic Process

n\rightarrow \infty

A RANDOM TRIADIC PROCESS

n→∞, the first homology group over

Extremal problems on shadows and hypercuts in simplicial complexes

\mathbb {Z}

Enumeration and randomized constructions of hypertrees

Z vanishes at the very moment when all the edges are covered by triangular faces.