Diszkrét matematika = Discrete mathematics

A pályázat résztvevői igen aktívak voltak a 2006-2008 években. Nemcsak sok eredményt értek el, miket több mint 150 cikkben publikáltak, eredményesen népszerűsítették azokat. Több mint 100 konferencián vettek részt és adtak elő, felerészben meghívott, vagy plenáris előadóként. Hagyományos gráfelmélet Több extremális gráfproblémát oldottunk meg. Új eredményeket kaptunk Ramsey számokról, globális és lokális kromatikus számokról, Hamiltonkörök létezéséséről. a crossig numberről, gráf kapacitásokról és kizárt részgráfokról. Véletlen gráfok, nagy gráfok, regularitási lemma Nagy gráfok "hasonlóságait" vizsgáltuk. Különféle metrikák ekvivalensek. Űj eredeményeink: Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit. Hipergráfok, egyéb kombinatorika Új Sperner tipusú tételekte kaptunk, aszimptotikusan meghatározva a halmazok max számát bizonyos kizárt struktőrák esetén. Több esetre megoldottuk a kizárt hipergráf problémát is. Elméleti számítástudomány Új ujjlenyomat kódokat és bioinformatikai eredményeket kaptunk. | The participants of the project were scientifically very active during the years 2006-2008. They did not only obtain many results, which are contained in their more than 150 papers appeared in strong journals, but effectively disseminated them in the scientific community. They participated and gave lectures in more than 100 conferences (with multiplicity), half of them were plenary or invited talks. Traditional graph theory Several extremal problems for graphs were solved. We obtained new results for certain Ramsey numbers, (local and global) chromatic numbers, existence of Hamiltonian cycles crossing numbers, graph capacities, and excluded subgraphs. Random graphs, large graphs, regularity lemma The "similarities" of large graphs were studied. We show that several different definitions of the metrics (and convergence) are equivalent. Several new results like the Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit were proved Hypergraphs, other combinatorics New Sperner type theorems were obtained, asymptotically determining the maximum number of sets in a family of subsets with certain excluded configurations. Several cases of the excluded hypergraph problem were solved. Theoretical computer science New fingerprint codes and results in bioinformatics were found

Regular Intersecting Families

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the same (or approximately the same) number of members of $\mathcal{F}$. In particular, we show that we can guarantee $|\mathcal{F}| = o({n-1\choose k-1})$ if and only if $k=o(n)$.Comment: 15 pages, accepted versio

Almost all triple systems with independent neighborhoods are semi-bipartite

The neighborhood of a pair of vertices $u,v$ in a triple system is the set of vertices $w$ such that $uvw$ is an edge. A triple system \HH is semi-bipartite if its vertex set contains a vertex subset $X$ such that every edge of \HH intersects $X$ in exactly two points. It is easy to see that if \HH is semi-bipartite, then the neighborhood of every pair of vertices in \HH is an independent set. We show a partial converse of this statement by proving that almost all triple systems with vertex sets $[n]$ and independent neighborhoods are semi-bipartite. Our result can be viewed as an extension of the Erd\H os-Kleitman-Rothschild theorem to triple systems. The proof uses the Frankl-R\"odl hypergraph regularity lemma, and stability theorems. Similar results have recently been proved for hypergraphs with various other local constraints

Mini-Workshop: Hypergraph Turan Problem

This mini-workshop focused on the hypergraph Turán problem. The interest in this difficult and old area was recently re-invigorated by many important developments such as the hypergraph regularity lemmas, flag algebras, and stability. The purpose of this meeting was to bring together experts in this field as well as promising young mathematicians to share expertise and initiate new collaborative projects

Regular intersecting families

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families F of k-element subsets of [n] := {1, ..., n}, such that every element of [n] lies in the same (or approximately the same) number of members of.F. In particular, we show that we can guarantee vertical bar vertical bar = o(((n-1)(k-1))) if and only if k = o(n). (C) 2019 Published by Elsevier B.V

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