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An Extremal Graph Problem with a Transcendental Solution

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  26 June 2018

DHRUV MUBAYI  and
CAROLINE TERRY
DHRUV MUBAYI
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607, USA (e-mail: mubayi@uic.edu)
CAROLINE TERRY
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (e-mail: cterry@umd.edu)
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Abstract

We prove that the number of multigraphs with vertex set {1, . . ., n} such that every four vertices span at most nine edges is an2+o(n2) where a is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy–Schwarz arguments.

Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

MSC classification

Primary: 05D99: None of the above, but in this section
Secondary: 05C35: Extremal problems 05C22: Signed and weighted graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 2 , March 2019 , pp. 303 - 324
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Research supported in part by NSF grant DMS 1300138.

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