On some edge Folkman numbers, small and large

Kaufmann, Jenny M.; Wickus, Henry J.; Radziszowski, Stanisław P.

doi:10.2140/involve.2019.12.813

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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

DOI: 10.2140/involve.2019.12.813

Abstract

Edge Folkman numbers $F_{e} (G_{1}, G_{2}; k)$ can be viewed as a generalization of more commonly studied Ramsey numbers. $F_{e} (G_{1}, G_{2}; k)$ 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 $F_{e} (J_{3}, K_{n}; n + 1) = 2 n - 1$ , $F_{e} (J_{3}, J_{n}; n) = 2 n - 1$ , and $F_{e} (J_{3}, J_{n}; n + 1) = 2 n - 3$ . Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on $F_{e} (J_{4}, J_{4}; k)$ for all $k > 4$ .

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Keywords

Folkman numbers, Ramsey numbers

Mathematical Subject Classification 2010

Primary: 05C55

Milestones

Received: 3 June 2018

Revised: 23 October 2018

Accepted: 29 November 2018

Published: 22 May 2019

Communicated by Kenneth S. Berenhaut

Authors

Jenny M. Kaufmann
Department of Mathematics Princeton University Princeton, NJ United States

Henry J. Wickus
Department of Mathematics and Computer Science DeSales University Center Valley, PA United States

Stanisław P. Radziszowski
Department of Computer Science Rochester Institute of Technology Rochester, NY United States

Vol. 12, No. 5, 2019