Summary: Graphs and Combinatorics 5, 307-314 (1989)
Sub-Ramsey Numbersfor Arithmetic Progressions
Noga Alon ~*, Yair Caro 1 and Zsolt Tuza 2.*
1 Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences,
Tel Aviv University, Ramat-Aviv, Tel Aviv 69978, Israel
2 Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest,
Kende u. 13-17, Hungary
Abstract. Let m >_3 and k > 1 be two given integers. A sub-k-coloring of [n] = (1,2,...,n} is
an assignment of colors to the numbers of [n] in which each color is used at most k times.
Call an S __qI-n] a rainbow set if no two of its elements have the same color. The sub-k-Ramsey
number sr(m,k) is defined as the minimum n such that every sub-k-coloring of [n] contains
a rainbow arithmetic progression of m terms. We prove that I2((k - 1)m2flogink) < sr(m,k) <
O((k - 1)m2log mk) as m ~ 0%and apply the same method to improve a previously known upper
bound for a problem concerning mappings from I-n]to I-n]without fixed points.
1. The Results
The aim of this note is to investigate certain Ramsey-type problems for arithmetic
progressions. Throughout, m-AP abbreviates "arithmetic progression of m terms."