On the order of countable graphs

Jaroslav Nešetřil; Saharon Shelah

doi:10.1016/S0195-6698(03)00064-7

On the order of countable graphs

Jaroslav Nešetřil^*, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

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

17 Scopus citations

Abstract

A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2^ω the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended ("partnership theorem"). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets.This is related to universal graphs, rigid graphs (where we solve a problem posed in J. Combin. Theory B 46 (1989) 133) and to the density problem for countable graphs.

Original language	English
Pages (from-to)	649-663
Number of pages	15
Journal	European Journal of Combinatorics
Volume	24
Issue number	6
DOIs	https://doi.org/10.1016/S0195-6698(03)00064-7
State	Published - Aug 2003

Keywords

Density
Partially ordered sets
Rigid graphs
Universal graphs

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10.1016/S0195-6698(03)00064-7

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On the order of countable graphs

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