Kombinatorikus módszerek a diszkrét geometriában = Combinatorial methods in discrete geometry

Barany Imre veletlen politopokkal, a Tverberg tetel altalanositasaival, a topologia es kombinatorikus geometria hatarteruletevel foglalkozott. Ugyanezen a teruleten dolgozott Kincses Janos, es meg politopok kombinatorikajan. Furedi Zoltan a ter illetve egy korlatos halmaz gazdasagos fedeseivel, grafok pakolasaival foglalkozott. Solymosi Jozsef additiv kombinatorikaval illetve additiv szamelmelettel foglalkozott, elsosorban azzal, hogy egy n elemu A szamhalmaz eseten legalabb mekkora az |A+A|+|A*A| ertek. Ezenkivul a Szemeredi-Trotter illetve Pach-Sharir tetelhez hasonlo incidencia tetelt bizonyotott magas dimenzios algebrai gorbekre. Ez utobbi eredmeny nagyon igeretes kezdetnek tunik. Toth Geza grafok metszesi szamaval es lerajzolasaival foglalkozott, konstualt egy grafot, amelynek a par-metszesi szama es paratlen-metszesi szama elter. Pach Janossal tanulmanyoztak grafok sikbeli es magasabb genuszu feluleten vett metszesi szamait, illetve az ezek kozti osszefuggeseket. A sik illetve a ter sokszoros fedeseinek szetbonthatosagat is vizsgalta. Pach Janossal azt vizsgaltak, hogy konvex halmazok rendtipusa mikor reprezentalhato pontokkal. Pach Janos Jacob Fox-szal a Lipton-Tarjan szeparator tetelt altalanositotta sikgrafokrol kulonbozo mas tipusu grafokra, peldaul konvex halmazok illetve gorbek metszetgrafjara. Por Attila lathatosagi grafokkal illetve grafok Kneser-reprezentaciojaval foglalkozott, amely szoros kapcsolatban van a frakcionalis kromatikus szammal. | Imre Barany investigated random polytopes, generalizations of the Tverberg theorem, and problems on the boundary of combinatorial geometry and topology. Janos Kincses also worked in this latter area, and also studied combinatorics of polytopes. Zoltan Furedi studied economical coverings of the space, or a bounded set. He also obtained important results concerning packings of small graphs into a large graph. Jozsef Solymosi worked in additive combinatorics and additive number theory. He investigated especially the question that at least how large is |A+A|+|A*A| if A is a set of n numbers. He also proved an incidence result for high dimensional algebraic curves, similar to the Szemeredi-Trotter or the Pach-Sharir theorems. This result seems to be a good start. Geza Toth investigated crossing numbers and drawings of graphs. He constructed a graph whose pair-crossing number is larger than its odd-crossing number. With Janos Pach he studied relationships between crossing numbers of graphs on different surfaces. He also obtained results on the decomposability of multiple coverings of the plane or space. With Janos Pach he studied, under what conditions can the order type of convex sets be represented by points. Janos Pach, together with Jacob Fox, generalized the Lipton-Tarjan separator theorem for planar graphs, for intersection graphs of convex sets, and for intersection graphs of curves. Attila Por obtained results on visibility graphs and on Kneser representations of graphs

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