Andrew J. Uzzell

The time of bootstrap percolation with dense initial sets for all thresholds

We study the percolation time of the r-neighbour bootstrap percolation model on the discrete torus i¾?/ni¾?d. For t at most a polylog function of n and initial infection probabilities within certain ranges depending on t, we prove that the percolation time of a random subset of the torus is exactly equal to t with high probability as n tends to infinity. Our proof rests crucially on three new extremal theorems that together establish an almost complete understanding of the geometric behaviour of the r-neighbour bootstrap process in the dense setting. The special case d-r=0 of our result was proved recently by Bollobas, Holmgren, Smith and Uzzell.

The time of bootstrap percolation with dense initial sets

Let r∈N . In r -neighbour bootstrap percolation on the vertex set of a graph G , vertices are initially infected independently with some probability p . At each time step, the infected set expands by infecting all uninfected vertices that have at least r infected neighbours. When p is close to 1, we study the distribution of the time at which all vertices become infected. Given t=t(n)=o(logn/loglogn) , we prove a sharp threshold result for the probability that percolation occurs by time t in d -neighbour bootstrap percolation on the d -dimensional discrete torus T d n . Moreover, we show that for certain ranges of p=p(n) , the time at which percolation occurs is concentrated either on a single value or on two consecutive values. We also prove corresponding results for the modified d -neighbour rule

On String Graph Limits and the Structure of a Typical String Graph

We study limits of convergent sequences of string graphs, that is, graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We also prove similar results for several related graph classes.

Multicolour containers, extremal entropy and counting

In breakthrough results, Saxton-Thomason and Balogh-Morris-Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton-Thomason and an entropy-based framework to deduce container and counting theorems for hereditary properties of k-colourings of very general objects, which include both vertex- and edge-colourings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterisation and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity and multicoloured graphs amongst others. Similar results were recently and independently obtained by Terry.

Strong Turán stability

Abstract We study the behaviour of K r + 1 -free graphs G of almost extremal size, that is, typically, e ( G ) = e x ( n , K r + 1 ) − O ( n ) . We show that such graphs must have a large amount of symmetry. In particular, if G is saturated, then all but very few of its vertices must have twins. As a corollary, we obtain a new proof of a theorem of Simonovits on the structure of extremal graphs with ω ( G ) ≤ r and χ ( G ) ≥ k for fixed k ≥ r ≥ 2 .

Regular colorings in regular graphs

Abstract An (r − 1, 1)-coloring of an r-regular graph G is an edge coloring (with arbitrarily many colors) such that each vertex is incident to r − 1 edges of one color and 1 edge of a different color. In this paper, we completely characterize all 4-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a (3, 1)-coloring. Also, for each r ≥ 6 we construct graphs that are not (r −1, 1)-colorable and, more generally, are not (r − t, t)-colorable for small t.