#### Vol. 12, No. 5, 2019

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
On some edge Folkman numbers, small and large

### Jenny M. Kaufmann, Henry J. Wickus and Stanisław P. Radziszowski

Vol. 12 (2019), No. 5, 813–822
##### Abstract

Edge Folkman numbers ${F}_{e}\left({G}_{1},{G}_{2};k\right)$ can be viewed as a generalization of more commonly studied Ramsey numbers. ${F}_{e}\left({G}_{1},{G}_{2};k\right)$ is defined as the smallest order of any ${K}_{k}$-free graph $F$ such that any red-blue coloring of the edges of $F$ contains either a red ${G}_{1}$ or a blue ${G}_{2}$. In this note, first we discuss edge Folkman numbers involving graphs ${J}_{s}={K}_{s}-e$, including the results ${F}_{e}\left({J}_{3},{K}_{n};n+1\right)=2n-1$, ${F}_{e}\left({J}_{3},{J}_{n};n\right)=2n-1$, and ${F}_{e}\left({J}_{3},{J}_{n};n+1\right)=2n-3$. Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on ${F}_{e}\left({J}_{4},{J}_{4};k\right)$ for all $k>4$.

##### Keywords
Folkman numbers, Ramsey numbers
Primary: 05C55