Infinite Ramsey-minimal graphs for star forests
Fawwaz Fakhrurrozi Hadiputra, Valentino Vito

TL;DR
This paper investigates the existence of Ramsey-minimal graphs for pairs of star forests, proving non-existence in cases involving infinite stars and constructing examples for finite cases, extending classical finite graph results to infinite graphs.
Contribution
It characterizes when Ramsey-minimal graphs exist for pairs of star forests, especially involving infinite stars, and constructs new examples for finite star forests, advancing the understanding of infinite Ramsey theory.
Findings
No Ramsey-minimal graphs exist for pairs with at least one infinite-star component.
Constructed a Ramsey-minimal graph for a finite star forest versus a subdivision graph.
Extended finite graph Ramsey results to the infinite case.
Abstract
For graphs , , and , we write if every red-blue coloring of the edges of produces a red copy of or a blue copy of . The graph is said to be -minimal if it is subgraph-minimal with respect to this property. The characterization problem for Ramsey-minimal graphs is classically done for finite graphs. In 2021, Barrett and the second author generalized this problem to infinite graphs. They asked which pairs admit a Ramsey-minimal graph and which ones do not. We show that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph. Also, we construct a Ramsey-minimal graph for a finite star forest versus a subdivision graph. This paper builds upon the results of Burr et al. in 1981 on Ramsey-minimal graphs for finite star forests.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Advanced Topology and Set Theory
