Tom Bohman

The early evolution of the H-free process

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as n→∞, the minimum degree in G is at least

Creating a Giant Component

cn^{1-(v_{H}-2)/(e_{H}-1)}(\log n)^{1/(e_{H}-1)}

Dynamic concentration of the triangle-free process

. This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs Kr,r with r≥5. When H is a complete graph Ks with s≥5 we show that for some C>0, with high probability the independence number of G is at most

How many random edges make a dense graph Hamiltonian

Cn^{2/(s+1)}(\log n)^{1-1/(e_{H}-1)}

Erdős–ko–rado in random hypergraphs

. This gives new lower bounds for Ramsey numbers R(s,t) for fixed s≥5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.

Adding random edges to dense graphs

Let

A sum packing problem of Erdös and the Conway-Guy sequence

c

Product rule wins A competitive game

be a constant and

SIR epidemics on random graphs with a fixed degree sequence

(e_1,f_1), (e_2,f_2), \dots, (e_{cn},f_{cn})

A nontrivial lower bound on the Shannon capacities of the complements of odd cycles

be a sequence of ordered pairs of edges on vertex set