Cut distance identifying graphon parameters over weak* limits

The theory of graphons comes with the so-called cut norm and the derived cut distance. The cut norm is finer than the weak* topology. Dole\v{z}al and Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin. Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a cut distance accumulation graphon can be pinpointed in the set of weak* accumulation points as a minimizer of the entropy. Motivated by this, we study graphon parameters with the property that their minimizers or maximizers identify cut distance accumulation points over the set of weak* accumulation points. We call such parameters cut distance identifying. Of particular importance are cut distance identifying parameters coming from subgraph densities, t(H,*). This concept is closely related to the emerging field of graph norms, and the notions of the step Sidorenko property and the step forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if and only if it is step Sidorenko, and that if a graph is norming then it is step forcing. Further, we study convexity properties of cut distance identifying graphon parameters, and find a way to identify cut distance limits using spectra of graphons. We also show that continuous cut distance identifying graphon parameters have the "pumping property", and thus can be used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming graphs, and a new section 3.2 on index pumping in the regularity lemm

The step Sidorenko property and non-norming edge-transitive graphs

Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a quasirandom multipartite graph minimizes the density of H among all graphs with the same edge densities between its parts; this property is called the step Sidorenko property. We show that many bipartite graphs fail to have the step Sidorenko property and use our results to show the existence of a bipartite edge-transitive graph that is not weakly norming; this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150].Comment: Minor correction on page

The step Sidorenko property and non-norming edge-transitive graphs

Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a quasirandom multipartite graph minimizes the density of H among all graphs with the same edge densities between its parts; this property is called the step Sidorenko property. We show that many bipartite graphs fail to have the step Sidorenko property and use our results to show the existence of a bipartite edge-transitive graph that is not weakly norming; this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150]

Relating the cut distance and the weak* topology for graphons

The theory of graphons is ultimately connected with the so-called cut norm. In this paper, we approach the cut norm topology via the weak* topology (when considering a predual of -functions). We prove that a sequence of graphons converges in the cut distance if and only if we have equality of the sets of weak* accumulation points and of weak* limit points of all sequences of graphons that are weakly isomorphic to . We further give a short descriptive set theoretic argument that each sequence of graphons contains a subsequence with the property above. This in particular provides an alternative proof of the theorem of Lovász and Szegedy about compactness of the space of graphons. We connect these results to “multiway cut” characterization of cut distance convergence from Borgs et al. (2012) [5]

Theory of combinatorial limits and extremal combinatorics

In the past years, techniques from different areas of mathematics have been successfully applied in extremal combinatorics problems. Examples include applications of number theory, geometry and group theory in Ramsey theory and analytical methods to different problems in extremal combinatorics. By providing an analytic point of view of many discrete problems, the theory of combinatorial limits led to substantial results in many areas of mathematics and computer science, in particular in extremal combinatorics. In this thesis, we explore the connection between combinatorial limits and extremal combinatorics. In particular, we prove that extremal graph theory problemsmay have unique optimal solutions with arbitrarily complex structure, study a property closely related to Sidorenko's conjecture, one of the most important open problems in extremal combinatorics, and prove a 30-year old conjecture of Gyori and Tuza regarding decomposing the edges of a graph into triangles and edges