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Induced Turán Numbers

Part of: Graph theory

Published online by Cambridge University Press:  02 November 2017

PO-SHEN LOH ,
MICHAEL TAIT ,
CRAIG TIMMONS  and
RODRIGO M. ZHOU
PO-SHEN LOH
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: ploh@cmu.edu, mtait@cmu.edu)
MICHAEL TAIT
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: ploh@cmu.edu, mtait@cmu.edu)
CRAIG TIMMONS
Affiliation:
Department of Mathematics and Statistics, California State University Sacramento, Sacramento, CA 95819, USA (e-mail: craig.timmons@csus.edu)
RODRIGO M. ZHOU
Affiliation:
COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21941-450, Brazil (e-mail: rzhou@cos.ufrj.br)
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Abstract

The classical Kővári–Sós–Turán theorem states that if G is an n-vertex graph with no copy of Ks,t as a subgraph, then the number of edges in G is at most O(n2−1/s). We prove that if one forbids Ks,t as an induced subgraph, and also forbids any fixed graph H as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in a Kr-free graph with no induced copy of Ks,t. This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C69: Dominating sets, independent sets, cliques
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 2 , March 2018 , pp. 274 - 288
Copyright
Copyright © Cambridge University Press 2017 

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