Rainbow Arithmetic Progressions and Anti-Ramsey Results

  • Authors:

  • Veselin Jungić;Jacob Licht;Mohammad Mahdian;Jaroslav Nešetřil;Radoš Radoičić

  • Affiliations:

  • Department of Mathematics, Simon Fraser University, Burnaby, BC, V54 1S6, Canada (e-mail: vjungic@sfu.ca);William H. Hall High School, 975 North Main Street, West Hartford, CT 06117, USA (e-mail: licht@mit.edu);Department of Mathematics, MIT, Cambridge, MA 02139, USA (e-mail: mahdian@math.mit.edu);Department of Applied Mathematics, Charles University, Prague, Czech Republic (e-mail: nesetril@kam.ms.mff.cuni.cz);Department of Mathematics, MIT, Cambridge, MA 02139, USA (e-mail: rados@math.mit.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2003

Quantified Score

Hi-index 0.00

Visualization

Abstract

The van der Waerden theorem in Ramsey theory states that, for every$k$ and $t$ and sufficiently large $N$, every $k$-colouring of$[N]$ contains a monochromatic arithmetic progression of length$t$. Motivated by this result, Radoičićconjectured that every equinumerous 3-colouring of $[3n]$ containsa 3-term rainbow arithmetic progression, i.e., an arithmeticprogression whose terms are coloured with distinct colours. In thispaper, we prove that every 3-colouring of the set of naturalnumbers for which each colour class has density more than 1/6,contains a 3-term rainbow arithmetic progression. We also provesimilar results for colourings of $\mathbb{Z}_n$. Finally, we givea general perspective on other anti-Ramsey-type problemsthat can be considered.