Endre Csóka

Invariant Gaussian processes and independent sets on regular graphs of large girth

We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. The best known bound is 0.4352n. In fact, computer simulation suggests that the bound our method provides is about 0.438n.

Limits of some combinatorial problems

Abstract We show some new examples how can limit theory help understanding combinatorial structures. We introduce two limit problems of Alperns Caching Game, which are good approximations of the original game when some parameters tend to infinity. With the use of these limit problems, we show some surprising results which radically changes our expectations about the structure of the optimal solution, e.g. this disproves the Kikuta-Ruckle Conjecture for Caching Games. For another example, we generalize the Manickam–Miklos–Singhi Conjecture, using limit theory.

On the graph limit question of Vera T. Sós

In the dense graph limit theory, the topology of the set of graphs is defined by the distribution of the subgraphs spanned by finite number of random vertices. Vera T. Sos proposed a question that if we consider only the number of edges in the spanned subgraphs, then whether it provides an equivalent definition. We show that the answer is positive on quasirandom graphs, and we prove a generalization of the statement.

A local flow algorithm in bounded degree networks

We show a deterministic local algorithm for constructing an almost maximum flow and an almost minimum cut in multisource-multitarget networks with bounded degrees. Locality means that we decide about each edge or node depending only on its constant radius neighbourhood. We show two applications of the flow algorithm, one is about how the neighborhood distributions of arbitrary bounded degree graphs can be approximated by bounded size graphs, and the other one is related to the Aldous-Lyons conjecture.