A partition theorem for a large dense linear order
| dc.creator | Dzamonja, M. | |
| dc.creator | Larson, J. | |
| dc.creator | Mitchell, W. | |
| dc.date | 2005-06-07 | |
| dc.date.accessioned | 2026-07-07T05:20:35Z | |
| dc.date.available | 2026-07-07T05:20:35Z | |
| dc.description | Let Q_K=(Q,<_Q)$ be a strongly K-dense linear order of size K for a suitable cardinal K. We prove, for all integers m > 1 that there is a finite value t_m^+ such that the set of all m-tuples from Q can be divided into t_m^+ many classes, such that whenever any of these classes C is colored with fewer than K many colors, there is a copy Q* of Q_K such that all m-tuples from Q* in C receive the same color. As a consequence we obtain that whenever we color the m-tuples of Q with fewer than K many colors, there is a copy of Q_K all m-tuples from which are colored in at most t_m^+ colors. In other words, the partition relation Q_K -->(Q_K)^m_{<K,r} holds for some finite r=t_m^+. We show that t_m^+ is the minimal value with this property. We were not able to give a formula for t_m^+ but we can describe t_m^+ as the cardinality of a certain finite set of types. We give an upper and a lower bound on its value and for m=2 we obtain t_2^+ = 2, while for m>2 we have t_m^+ > t_m, the m-th tangent number. The paper also contains similar partition results about K-Rado graphs. A consequence of our work and some earlier results of Hajnal and Komjath is that a theorem of Shelah known to follow from a large cardinal assumption in a generic extension, does not follow from any large cardinal assumption on its own. | |
| dc.description | LaTeX, 77 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/math/0506123 | |
| dc.identifier | http://arxiv.org/abs/math/0506123 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75428 | |
| dc.subject | Logic | |
| dc.subject | 03E02 (primary), 03E35 (secondary) | |
| dc.title | A partition theorem for a large dense linear order | |
| dc.type | text |