A counterexample to a geometric Hales-Jewett type conjecture

Authors

  • Vytautas Gruslys University of Cambridge

DOI:

https://doi.org/10.20382/jocg.v5i1a11

Abstract

Pór and Wood conjectured that for all \(k,l \ge 2\) there exists \(n \ge 2\) with the following property: whenever \(n\) points, no \(l + 1\) of which are collinear, are chosen in the plane and each of them is assigned one of \(k\) colours, then there must be a line (that is, a maximal set of collinear points) all of whose points have the same colour. The conjecture is easily seen to be true for \(l = 2\) (by the pigeonhole principle) and in the case \(k = 2\) it is an immediate corollary of the Motzkin-Rabin theorem. In this note we show that the conjecture is false for \(k, l \ge 3\).

Downloads

Download data is not yet available.

Author Biography

Vytautas Gruslys, University of Cambridge

Research student

Downloads

Published

2014-10-20

Issue

Vol. 5 No. 1 (2014)

Section

Articles

License

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).