Universal graphs omitting finitely many finite graphs

Péter Komjáth; Saharon Shelah

doi:10.1016/j.disc.2019.111596

Universal graphs omitting finitely many finite graphs

Péter Komjáth^*
, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

If F is a family of graphs, then a graph is F-free, if it contains no induced subgraph isomorphic to an element of F. If F is a finite set of finite graphs, λ is an infinite cardinal, we let CF(F,λ) be the minimal number of F-free graphs of size λ such that each F-free graph of size λ embeds into some of them. We show that if 2^<λ=λ, then CF(F,λ)≤c (continuum), there are examples such that CF(F,λ) is finite but can be arbitrarily large, and give an example such that CF(F,λ)≥c for any infinite cardinal λ.

Original language	English
Article number	111596
Journal	Discrete Mathematics
Volume	342
Issue number	12
DOIs	https://doi.org/10.1016/j.disc.2019.111596
State	Published - Dec 2019

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Publisher Copyright:
© 2019

Keywords

Infinite graphs
Universal graphs

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10.1016/j.disc.2019.111596

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abstract = "If F is a family of graphs, then a graph is F-free, if it contains no induced subgraph isomorphic to an element of F. If F is a finite set of finite graphs, λ is an infinite cardinal, we let CF(F,λ) be the minimal number of F-free graphs of size λ such that each F-free graph of size λ embeds into some of them. We show that if 2<λ=λ, then CF(F,λ)≤c (continuum), there are examples such that CF(F,λ) is finite but can be arbitrarily large, and give an example such that CF(F,λ)≥c for any infinite cardinal λ.",

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Universal graphs omitting finitely many finite graphs

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