Owings-like theorems for infinitely many colours or finite monochromatic sets

• Published in: Annals of pure and applied logic Vol. 175; no. 10; p. 103495
• Main Authors: Fernández-Bretón, David J., Sarmiento Rosales, Eliseo, Vera, Germán
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-12-2024
• Subjects: Abelian group; Additive combinatorics; Combinatorial set theory; Owings's problem; Ramsey-type theorems; secondary; Abelian group; Additive combinatorics; Owings's problem; Combinatorial set theory; Ramsey-type theorems; primary
• ISSN: 0168-0072
• DOI: 10.1016/j.apal.2024.103495

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers κ,θ, every colouring c:G⟶θ yields a subset X⊆G with |X|=κ such that X+X is monochromatic. (Owings's problem asks this for G=Z, θ=2 and κ=ℵ0; this is known to be false for the same G and κ but θ=3.) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.