2024/09/19 by L. Boza, Boza, Luis
Computer Science · Mathematics · #05C55 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #G.2.2 #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2409.12770
openalex publication_date 2024/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study Ramsey numbers of the form R(C4,K1,n). We determine the eight previously unknown values of R(C4,K1,n) for n≤ 38. In particular, we show that R(C4,K1,27)=33 and R(C4,K1,n)=n+7 for 28≤ n≤ 33 and for n=37. We also establish new general inequalities relating different values of this function. Specifically, if m≡ 2\pmod 6 with m≥ 8, then R(C4,K1,m2+3)≤ m2+m+4, and for all positive integers a and b, either R(C4,K1,a)≥ a+b or R(C4,K1,a+b)≤ a+2b. As consequences, we obtain the functional inequalities f(2n-f(n)+1)≥ n and f(f(n)+1)≤ 2f(n)-n+2, where f(n)=R(C4,K1,n).