Geodesic
Ramsey numbers with prescribed rate of growth
The electronic journal of combinatorics, Tome 30 (2023) no. 3
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Let $R(G)$ be the $2$-colour Ramsey number of a graph $G$.In this note, we prove that for any non-decreasing function $n \leq f(n) \leq R(K_n)$, there exists a sequence of connected graphs $(G_n)_{n\in\mathbb N}$, with $|V(G_n)| = n$ for all $n \geq 1$, such that $R(G_n) = \Theta(f(n))$. In contrast, we also show that an analogous statement does not hold for hypergraphs of uniformity at least $5$.We also use our techniques to answer in the affirmative a question posed by DeBiasio about the existence of sequences of graphs, but whose $2$-colour Ramsey number is linear whereas their $3$-colour Ramsey number has superlinear growth.
DOI : 10.37236/11652
Classification : 05C55, 05D10, 05C65
Mots-clés : 2-colour Ramsey number

Matías Pavez-Signé    ; Simón Piga  1   ; Nicolás Sanhueza-Matamala 

1 University of Birmingham 
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     author = {Mat{\'\i}as  Pavez-Sign\'e and Sim\'on Piga and Nicol\'as  Sanhueza-Matamala},
     title = {Ramsey numbers with prescribed rate of growth},
     journal = {The electronic journal of combinatorics},
     year = {2023},
     volume = {30},
     number = {3},
     doi = {10.37236/11652},
     zbl = {1533.05185},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/11652/}
}
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Matías  Pavez-Signé; Simón Piga; Nicolás  Sanhueza-Matamala. Ramsey numbers with prescribed rate of growth. The electronic journal of combinatorics, Tome 30 (2023) no. 3. doi: 10.37236/11652

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