***Peter Csikvari: On Sidorenko’s conjecture for determinants and Gaussian Markov random fields***
We study a class of determinant inequalities that are closely related to Sidorenko’s famous conjecture (Also conjectured by Erdős and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for Gaussian Markov random fields (GMRF). We call a GMRF on a finite graph G homogeneous if the marginal distributions on the edges are all identical. We show that if G is bipartite then the differential entropy of any homogeneous GMRF on G is at least |E(G)| times the edge entropy plus |V (G)| − 2|E(G)| times the point entropy. We also show that in the case of non-negative correlation on edges, the result holds for an arbitrary graph G. The connection between Sidorenko’s conjecture and GMRF’s is established via a large deviation principle on high dimensional spheres combined with graph limit theory. Connection with Ihara zeta function and the number of spanning trees is also discussed. Joint work with Balázs Szegedy.
***Ewan Davies: Tight bounds on the coefficients of partition functions via stability***
We show how stability results follow naturally from the recently developed
occupancy method for maximising and minimising physical observables over
classes of regular graphs, and then show these stability results can be used to
obtain tight extremal bounds on the individual coefficients of the corresponding
partition functions. As applications, we prove new bounds on the number of
independent sets and matchings of a given size in regular graphs.
For large enough graphs and almost all sizes, the bounds are tight and confirm
the Upper Matching Conjecture of Friedland, Krop, and Markström, and a
conjecture of Kahn on independent sets for a wide range of parameters.
***Martin Dolezal: Convergence of graphons and structuredness order***
We show that compactness of the cut-distance follows from compactness of the Vietoris topology of the hyperspace over functions equipped with the weak* topology. We introduce a new order on the space of graphons which naturally emerges from these proofs. This order allows to compare how structured two graphons are.
***Christos Pelekis: A continuous analogue of Sperner's theorem***
A Sperner-set is a subset of the unit square that does not contain two elements x and y such that x is dominating y coordinate-wise. I will show that the Hausdorff dimension of a Sperner set is at most 1, that the 1-dimensional Hausdorff measure of a Sperner-set is at most 2 and that both results are sharp. If time permits, I will briefly discuss continuous analogues of the Erdös-Ko-Rado theorem and the de Bruijn-Erdös theorem. This is joint work with Martin Doležal, Konrad Engel and Themis Mitsis.
***Diana Piguet: Tilings in graphons***
***Israel Rocha: Perfect graphons: colorings and cliques***
While turning a graph definition into a graphon one is typically straightforward, counterparts to many natural facts from finite graphs turned out to be quite challenging. We introduce graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. Turns out these parameters are not continuous with respect to the cut-norm. We show certain relations between these parameters and introduce a notion of perfect graphons, characterized in terms of induced densities of odd cycles and its complements.
***Matas Sileikis: Limit distribution of clique counts in dense inhomogeneous graphs***
Given a graphon W, we sample uniform independent points x_1, ..., x_n
from [0,1] and connect x_i and x_j with probability W(x_i,x_j). This
gives a random (dense) graph model for which we study the basic
statistic: the number X_H of copies of a fixed graph H. In the talk I
will describe the limit distribution of X_H when H is a clique and
characterize when this limit is normal. (Joint work with J. Hladky and
C. Pelekis)
***Fan Wei: Local Approach to Sidorenko's conjecture***
A famous conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if $H$ is a bipartite graph, then the random graph with edge density $p$ has in expectation asymptotically the minimum number of copies of $H$ over all graphs of the same number of vertices and edge density. Lov\'asz proved a local version of this conjecture. Here we extend this result, proving that the local version holds for a graph if and only if the graph is a forest or it has even girth.
We will also discuss some results for the local approach for Ramsey multiplicity problem, which is still widely open.