Taiwanese Journal of Mathematics

On Generalized Folkman Numbers

Yusheng Li and Qizhong Lin

Full-text: Open access

Abstract

For graphs $G$, $G_1$ and $G_2$, let $G \to (G_1,G_2)$ signify that any red/blue edge-coloring of $G$ contains a red $G_1$ or a blue $G_2$, and let $f(G_1,G_2)$ be the minimum $N$ such that there is a graph $G$ of order $N$ with $\omega(G) = \max \{\omega(G_1),\omega(G_2)\}$ and $G \to (G_1,G_2)$. It is shown that $c_1(n/\!\log n)^{(m+1)/2} \leq f(K_m,K_{n,n}) \leq c_2 n^{m-1}$, where $c_i = c_i(m) \gt 0$ are constants. In particular, $cn^2/\log n \leq f(K_3,K_{n,n}) \leq 2n^2+2n-1$. Moreover, $f(K_m,T_n) \leq m^2(n-1)$ for all $n \geq m \geq 2$, where $T_n$ is a tree on $n$ vertices.

Article information

Source
Taiwanese J. Math., Volume 21, Number 1 (2017), 1-9.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874553

Digital Object Identifier
doi:10.11650/tjm.21.2017.7710

Mathematical Reviews number (MathSciNet)
MR3613970

Zentralblatt MATH identifier
1357.05045

Subjects
Primary: 05C35: Extremal problems [See also 90C35] 05C55: Generalized Ramsey theory [See also 05D10] 05D10: Ramsey theory [See also 05C55]

Keywords
generalized Folkman number construction probabilistic method

Citation

Li, Yusheng; Lin, Qizhong. On Generalized Folkman Numbers. Taiwanese J. Math. 21 (2017), no. 1, 1--9. doi:10.11650/tjm.21.2017.7710. https://projecteuclid.org/euclid.twjm/1498874553


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