Continuous versions of some extremal hypergraph problems. II

“…Thus (C(t, O))), which contradicts (12). Since |O| < n we may apply the induction hypothesis for π O (v), (A(t, O)) instead of v, n, k, ε, d 1 , .…”

Section: Lemma 42mentioning

confidence: 89%

A continuous analogue of Erdős' k-Sperner theorem

Mitsis

Pelekis

Vlasák

2020

Journal of Mathematical Analysis and Applications

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“…Already in the 1970's it was realized (see [2,10,11,12]) that several results from extremal graph theory have measure-theoretic counterparts. In this setting, one is interested in the maximum "number of edges" in measure graphs, i.e., graphs whose vertex set corresponds to some measure space X and whose edge set corresponds to a symmetric subset of the product space that does not intersect the diagonal, i.e., it contains no points of the form (x, x), where x ∈ X.…”

Section: Large-distance Graphsmentioning

confidence: 99%

A Turán-Type Theorem for Large-Distance Graphs in Euclidean Spaces, and Related Isodiametric Problems

Doležal

Hladký

Kolář

et al. 2020

Discrete Comput Geom

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Given a measurable set A ⊂ R d we consider the large-distance graph G A , on the ground set A, in which each pair of points from A whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the 2d-dimensional Lebesgue measure of the edge set as well as the d-dimensional Lebesgue measure of the vertex set of a large-distance graph in the d-dimensional Euclidean space that contains no copies of a complete graph on k vertices. The former problem may be seen as a continuous analogue of Turán's classical graph theorem, and the latter as a "graph-theoretic" analogue of the classical isodiametric problem. Our main result yields an analogue of Mantel's theorem for large-distance graphs. Our approach employs an isodiametric inequality in an annulus, which might be of independent interest.Keywords Turán's theorem · isodiametric problem · distance graph Mathematics Subject Classification (2010) 05C63 · 51K99 · 05C35 · 51M16 1 Prologue, related work and main resultsLet us begin with a folklore result of Turán which pertains to graphs that contain no complete graph (also called a clique) on k vertices. Given a graph G = (V, E), we denote by |V | and |E| the number of its vertices and edges, respectively. Theorem 1.1 (Turán [22]) Let G = (V, E) be a graph which does not contain a complete graph on k vertices. Then |E| ≤ 1 2 1 − 1 k−1 · |V | 2 .

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Continuous versions of some extremal hypergraph problems. II

Cited by 18 publications

References 8 publications

Turán’s graph theorem, measures and probability theory

Turán’s graph theorem, measures and probability theory

A continuous analogue of Erdős' k-Sperner theorem

A Turán-Type Theorem for Large-Distance Graphs in Euclidean Spaces, and Related Isodiametric Problems

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