On the Fon-der-Flaass Interpretation of Extremal Examples for Turan's (3,4)-problem

In 1941, Turan conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many other examples of this density. Fon-der-Flaass (1988) presented a general construction that converts an arbitrary $\vec C_4$-free orgraph $\Gamma$ into a Turan (3,4)-graph. He observed that all Turan-Brown-Kostochka examples result from his construction, and proved the bound >= 3/7(1-o(1)) on the edge density of any Turan (3,4)-graph obtainable in this way. In this paper we establish the optimal bound 4/9(1-o(1)) on the edge density of any Turan (3,4)-graph resulting from the Fon-der-Flaass construction under any of the following assumptions on the undirected graph $G$ underlying the orgraph $\Gamma$: 1. $G$ is complete multipartite; 2. The edge density of $G$ is >= (2/3-epsilon) for some absolute constant epsilon>0. We are also able to improve Fon-der-Flaass's bound to 7/16(1-o(1)) without any extra assumptions on $\Gamma$

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples

Graph Theory

This was a workshop on graph theory, with a comprehensive approach. Highlights included the emerging theories of sparse graph limits and of infinite matroids, new techniques for colouring graphs on surfaces, and extensions of graph minor theory to directed graphs and to immersion

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Approximate equivalence relations

Generalizing results for approximate subgroups, we study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover’s normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim–Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations. We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov–Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric. For sequences of approximate equivalence relations with an “approximately unique” probability logic 1-type, we obtain a structure theorem generalizing the “Lie model” theorem for approximate subgroups (Theorem 5.5). The models here are Riemannian homogeneous spaces, fibered over a locally finite graph. Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin. This generalizes the main lemma for strong approximation of groups. For NIP theories, pursuing a question of Pillay’s, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets. This can be seen as an archimedean analogue of results of Macpherson and Tent on NIP profinite groups