Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures

AbstractWe consider extremal problems ‘of Turán type’ for r-uniform ordered hypergraphs, where multiple oriented edges are permitted up to multiplicity q. With any such ‘(r, q)-graph’ Gn we associate an r-linear form whose maximum over the standard (n − 1)-simplex in Rn is called the (graph-) density g(Gn) of Gn. If ex(n, L) is the maximum number of oriented hyperedges in an n-vertex (r, q)-graph not containing a member of L, limn→∞ ex(n, L)/nr is called the extremal density of L. Motivated, in part, from results for ordinary graphs, digraphs, and multigraphs, we establish relations between these two notions

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

The maximal length of a gap between r-graph Tur\'an densities

The Tur\'an density $\pi(\cal F)$ of a family $\cal F$ of $r$-graphs is the limit as $n\to\infty$ of the maximum edge density of an $\cal F$-free $r$-graph on $n$ vertices. Erdos [Israel J. Math 2 (1964) 183--190] proved that no Tur\'an density can lie in the open interval $(0,r!/r^r)$. Here we show that any other open subinterval of $[0,1]$ avoiding Tur\'an densities has strictly smaller length. In particular, this implies a conjecture of Grosu [E-print arXiv:1403.4653v1, 2014].Comment: 7 page

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