Another note on intervals in the Hales-Jewett theorem
- The Hales-Jewett Theorem states that any r-colouring of [m]ⁿ contains a monochromatic combinatorial line if n is large enough. Shelah's proof of the theorem implies that for m = 3 there always exists a monochromatic combinatorial line whose set of active coordinates is the union of at most r intervals. For odd r, Conlon and Kamčev constructed r–colourings for which it cannot be fewer than r intervals. However, we show that for even r and large n, any r–colouring of [3]ⁿ contains a monochromatic combinatorial line whose set of active coordinates is the union of at most r−1 intervals. This is optimal and extends a result of Leader and Räty for r=2.
| Author: | Nina Kamčev, Christoph Spiegel |
|---|---|
| Document Type: | Article |
| Parent Title (English): | Electronic Journal of Combinatorics |
| Volume: | 29 |
| Issue: | 1 |
| First Page: | P1.62 |
| Year of first publication: | 2022 |
| DOI: | https://doi.org/10.37236/6730 |
