Norm-Graphs: Variations and Applications

Noga Alon*, Lajos Rónyai, Tibor Szabó

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We describe several variants of the norm-graphs introduced by Kollár, Rónyai, and Szabó and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 12n5/3 edges, containing no copy of K3, 3, thus slightly improving an old construction of Brown. We also prove that the maximum number of vertices in a complete graph whose edges can be colored by k colors with no monochromatic copy of K3, 3 is (1+o(1))k3. This answers a question of Chung and Graham. In addition we prove that for every fixed t, there is a family of subsets of an n element set whose so-called dual shatter function is O(mt) and whose discrepancy is Ω(n1/2-1/2tlogn). This settles a problem of Matoušek.

Original languageEnglish
Pages (from-to)280-290
Number of pages11
JournalJournal of Combinatorial Theory. Series B
Volume76
Issue number2
DOIs
StatePublished - Jul 1999

Funding

FundersFunder number
Hermann Minkowski Minerva Center for Geometry
NWO-OTKA048.011.002, FKFP 0612 1997
Alfred P. Sloan Foundation96-6-2
Bloom's Syndrome Foundation
State of New Jersey Department of State
European Commission
Hungarian Scientific Research Fund016524, 016503
Tel Aviv University

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