Manual Combinatorial geometry and its algorithmic applications

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combinatorial optimization

In DeMarco and Kahn conjectured the correct order as n grows of the logarithm of the probability in question. Their guess was confirmed in a series of papers assuming p tends to zero not too fast. September, There are numerous models of random rooted trees and many of them can be represented as the genealogical tree of the critical Galton-Watson process, conditioned to 'die out' after n individuals come to life.

June, The theory of graphons naturally comes with the so-called cut distance. The key result, which makes the theory applicable, is that graphons together with the cut distance topology form a compact space. This was first proven in the seminal paper by Lovasz and Szegedy in The relation of the cut distance topology and various norms most notably Lp-norms has been studied extensively in the past.

In the current paper , Israel Rocha and Vaclav Rozhon together with their collaborators Martin Dolezal, Jan Grebik and Jan Hladky thoroughly investigate the relation between the cut distance and the weak-star topology. They further give a short descriptive set theoretic argument that each sequence of graphons contains a subsequence with the property above, thus reproving the Lovasz-Szegedy theorem.

There will be a follow-up paper, where this theory is linked to graph on parameters, including graph norms. So, stay tuned! The work he presented at the competition was done within the project Extremal graph theory and applications of the Czech Science Foundation and among other includes the following two preprints: arXiv April, Turan-type questions are among the most central in extremal graph theory. In that setting, the task is to find density conditions on the host graph that guarantee the containment of a given graph F. Famous conjectures of Erdos and Sos from and of Loebl, Komlos and Sos from the latter one solved asymptotically in [ 1 , 2 , 3 , 4 ] give such conditions when F is a tree.

We wish him all the best for his studies abroad. November, The concept of a uniform spanning tree UST of a graph plays a central role in probability theory and mathematical aspects of statistical physics. The main reason for this prominence is the connection to the Schramm-Loewner evolution through loop-erased random walks.

The award of the Fields medal to Wendelin Werner was largely on work on this connection. While the probability community typically investigates USTs of bounded-degree graphs such as large planar grids, Jan Hladky and Tuan Tran, who were both members of ExtrA when the project was initiated, together with Asaf Nachmias of Tel Aviv University looked at the area from the perspective of limits of dense graphs.

They describe the local structure of a typical UST of any graph that is close to a given graphon. The paper will appear in Journal of Statistical Physics. November, Many of the most important problems in extremal graph theory concern graph packings. In that setting, the task is to find copies of several given graphs into one host graph, so that the copies are edge-disjoint. There has been a flurry of research in the area of graph packing recently including [1] , [2] , [3] , [4] , [5] , mostly driven by old Tree Packing conjectures of Ringel and Gyarfas and Lehel.

They prove that any family of graphs of bounded degeneracy with a moderate bound on the maximum degree packs into a complete graph, subject to the obvious condition on not "overpacking". July, Christos Pelekis introduces a Hausdorf dimension point of view on problems in extremal combinatorics.