Abstract: Many interacting particle systems can in a natural way be constructed from graphical representations. The latter are basically collections of independent Poisson processes that describe at which times certain local maps should be applied to the system. If all maps are additive, this naturally leads to a rather simple and elegant duality. More generally, if the maps are monotone but not necessarily additive, there is still a dual process, but of a more complicated nature. We study this dual process for a particle system with cooperative branching and deaths on the complete graph with N vertices. In the limit as N tends to infinity, this leads to a Markov process taking values in hypergraphs, whose long-time behavior can largely be solved.

Abstract: Komlos determined the asymptotically optimal minimum degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H. We show that the minimum degree condition can be relaxed in the sense that we require only a given fraction of vertices to have the prescribed degree. Joint work with Diana Piguet.

Abstract: Given a fixed graph H and natural numbers n and b, the strict (1:b) Avoider-Enforcer H-game, played on the edge set of K_n, has the following rules: In each turn Avoider picks exactly one edge, and then Enforcer picks exactly b edges. Avoider wins if and only if his graph is H-free after all edges are taken. The lower threshold is the largest b_0 for which Enforcer has a winning strategy for the (1:b) H-game played on K_n for any b \leq b_0, and the upper threshold is the largest b for which Enforcer wins the (1:b) game.
The separation conjecture of Hefetz, Krivelevich, Stojakovic and Szabo states that for any non-trivial connected graph H, the lower threshold and the upper threshold of the Avoider-Enforcer H-game played on K_n are not of the same order in n. Until now, the conjecture has been verified only for stars, by Grzesik, Mikalacki, Nagy, Naor, Patkos and Skerman. In this work, we show that the conjecture holds for any unicyclic graph H.
Joint work with Bednarska-Bzdega, Ben-Eliezer and Gishboliner.

Abstract: Graphons are the main objects in the theory of graph limits introduced in 2006 by Lovász and Szegedy, and recently developed by Borgs, Chayes, Lovasz, Sos and Vesztergombi. Graphons are limits of graph sequences in the cut metric. Graphs are particular types of graphons and the space of graphons is compact with respect the cut metric. This is equivalent to strong forms of Szemeredi’s Regularity Lemma. That implies that the space of graphons is indeed the completion of the space of finite graphs with the cut metric. Naturally, one wants to extend the classical graph parameters to graphons and hope to find continuity. Even though some parameters are continuous, as the density of triangles, some are not, as the matching number. Whenever we do not have continuity, properties do not follow directly from graphs. We introduce graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the b-fractional clique number. Turns out these parameters are not continuous. In this talk I will explain how this can be done, the properties we can obtain, and the significant differences between the proofs for graphs and for graphons.

Abstract: Komlós conjectured in 1981 that among all graphs with minimum degree at least d, the complete graph K_{d+1} minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when d is sufficiently large. In fact we prove a stronger result: for large d, any graph G with average degree at least d contains almost twice as many Hamiltonian subsets as K_{d+1}, unless G is isomorphic to K_{d+1} or a certain other graph which we specify.
This is joint work with Jaehoon Kim, Hong Liu and Katherine Staden.

Abstract:Bootstrap percolation is a simple process which is used as a model of propagation phenomena in real world networks including, for example, the spread of a rumour in a social network, the dynamics of ferromagnetism and information processing in neural networks. Given a graph G and an integer r, the r-neighbour bootstrap process begins with a set of "infected" vertices and, in each step, a "healthy" vertex becomes infected if it has at least r infected neighbours. A central problem in the area is to determine the size of the smallest initial infection which will spread to every vertex of the graph. In this talk, I will present a new trick for obtaining lower bounds on this quantity by transforming the problem into an infection problem on the edges of the graph and applying some basic facts from linear algebra. In particular, I will outline a proof of a conjecture of Balogh and Bollobás (2006) on the smallest infection which spreads to every vertex of a high-dimensional square lattice and mention some potential applications to analysing the behaviour of a random infection in this setting. This talk is based on joint work with Natasha Morrison.

Abstract: According to the isodiametric problem the area of a set whose diameter is 2 cannot be larger than π. What is the maximum area of a planar set, A, with the property that among any three points in A at least two are at distance less than or equal to 2? I will discuss the, devious, motivation behind this question and I will sketch the proofs of certain results and bounds on the maximum area of A.