156 research outputs found

    Chromatic number of graphs and edge Folkman numbers

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    In the paper we give a lower bound for the number of vertices of a given graph using its chromatic number. We find the graphs for which this bound is exact. The results are applied in the theory of Foklman numbers.Comment: 9 pages, 1 figur

    HipergrĂĄfok = Hypergraphs

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    A projekt cĂ©lkitƱzĂ©seit sikerĂŒlt megvalĂłsĂ­tani. A nĂ©gy Ă©v sorĂĄn több mint szĂĄz kivĂĄlĂł eredmĂ©ny szĂŒletett, amibƑl eddig 84 dolgozat jelent meg a tĂ©ma legkivĂĄlĂłbb folyĂłirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. SzĂĄmos rĂ©gĂłta fennĂĄllĂł sejtĂ©st bebizonyĂ­tottunk, egĂ©sz rĂ©gi nyitott problĂ©mĂĄt megoldottunk hipergrĂĄfokkal kapcsolatban illetve kapcsolĂłdĂł terĂŒleteken. A problĂ©mĂĄk nĂ©melyike sok Ă©ve, olykor több Ă©vtizede nyitott volt. Nem egy közvetlen kutatĂĄsi eredmĂ©ny, de szintĂ©n bizonyos Ă©rtĂ©kmĂ©rƑ, hogy a rĂ©sztvevƑk egyike a NorvĂ©g KirĂĄlyi AkadĂ©mia tagja lett Ă©s elnyerte a Steele dĂ­jat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize

    Diszkrét matematika = Discrete mathematics

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    A pĂĄlyĂĄzat rĂ©sztvevƑi igen aktĂ­vak voltak a 2006-2008 Ă©vekben. Nemcsak sok eredmĂ©nyt Ă©rtek el, miket több mint 150 cikkben publikĂĄltak, eredmĂ©nyesen nĂ©pszerƱsĂ­tettĂ©k azokat. Több mint 100 konferenciĂĄn vettek rĂ©szt Ă©s adtak elƑ, felerĂ©szben meghĂ­vott, vagy plenĂĄris elƑadĂłkĂ©nt. HagyomĂĄnyos grĂĄfelmĂ©let Több extremĂĄlis grĂĄfproblĂ©mĂĄt oldottunk meg. Új eredmĂ©nyeket kaptunk Ramsey szĂĄmokrĂłl, globĂĄlis Ă©s lokĂĄlis kromatikus szĂĄmokrĂłl, Hamiltonkörök lĂ©tezĂ©sĂ©sĂ©rƑl. a crossig numberrƑl, grĂĄf kapacitĂĄsokrĂłl Ă©s kizĂĄrt rĂ©szgrĂĄfokrĂłl. VĂ©letlen grĂĄfok, nagy grĂĄfok, regularitĂĄsi lemma Nagy grĂĄfok "hasonlĂłsĂĄgait" vizsgĂĄltuk. KĂŒlönfĂ©le metrikĂĄk ekvivalensek. Ć°j eredemĂ©nyeink: Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit. HipergrĂĄfok, egyĂ©b kombinatorika Új Sperner tipusĂș tĂ©telekte kaptunk, aszimptotikusan meghatĂĄrozva a halmazok max szĂĄmĂĄt bizonyos kizĂĄrt struktƑrĂĄk esetĂ©n. Több esetre megoldottuk a kizĂĄrt hipergrĂĄf problĂ©mĂĄt is. ElmĂ©leti szĂĄmĂ­tĂĄstudomĂĄny Új ujjlenyomat kĂłdokat Ă©s bioinformatikai eredmĂ©nyeket 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

    Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces

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    Two applications of Nash-Williams' theory of barriers to sequences on Banach spaces are presented: The first one is the c0c_0-saturation of C(K)C(K), KK countable compacta. The second one is the construction of weakly-null sequences generalizing the example of Maurey-Rosenthal

    PPP-Completeness and Extremal Combinatorics

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