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Extremální vlastnosti hypergrafů / Extremální vlastnosti hypergrafů

We give an overview of recent progress in the research of hypergraph jumps -- a problem from extremal combinatorics. The number $\alpha \in [0, 1)$ is a jump for $r$ if for any $\epsilon > 0$ and any integer $m \ge r$ any $r$-graph with $N > N(\epsilon, m)$ vertices and at least $(\alpha + \epsilon) {N \choose r}$ edges contains a subgraph with $m$ vertices and at least $(\alpha + c) {m \choose r}$ edges, where $c := c(\alpha)$ does depend only on $\alpha$. Baber and Talbot \cite{Baber} recently gave first examples of jumps for $r = 3$ in the interval $[2/9, 1)$. Their result uses the framework of flag algebras \cite{Raz07} and involves solving a semidefinite optimization problem. A software implementation of their method is a part of this work.

Identiferoai:union.ndltd.org:nusl.cz/oai:invenio.nusl.cz:313900
Date January 2011
CreatorsMach, Lukáš
ContributorsKráľ, Daniel, Kaiser, Tomáš
Source SetsCzech ETDs
LanguageEnglish
Detected LanguageEnglish
Typeinfo:eu-repo/semantics/masterThesis
Rightsinfo:eu-repo/semantics/restrictedAccess

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