Indicators, Chains, Antichains, Ramsey Property

Sokić, Miodrag

Canadian mathematical bulletin, Tome 57 (2014) no. 3, pp. 631-639

Résumé

We introduce two Ramsey classes of finite relational structures. The first class contains finite structures of the form $\left( A,\,\left( {{I}_{i}} \right)_{i=1}^{n},\le ,\left( {{\underline{\prec }}_{i}} \right)_{i=1}^{n} \right)$ , where $\le$ is a total ordering on $A$ and ${{\underline{\prec }}_{i}}$ is a linear ordering on the set $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$ . The second class contains structures of the form a $\left( a,\le ,\left( {{i}_{i}} \right)_{i=1}^{n},\underline{\prec } \right)$ , where $\left( A,\,\le\right)$ is a weak ordering and $\underline{\prec }$ is a linear ordering on $A$ such that $A$ is partitioned by $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$ into maximal chains in the partial ordering $\le$ and each $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$ is an interval with respect to $\underline{\prec }$ .

Détail

DOI : 10.4153/CMB-2013-028-0

Mots-clés : 05C55, 03C15, 54H20, Ramsey property, linear orderings

Bibliographie

[1] [1] Abramson, F. G., Harrington, L. A., Models without indiscernibles. J. Symbolic Logic 43 (1978), no. 3, 572–600. Google Scholar | DOI

[2] [2] Graham, R. L., Rotschild, B. L., and Spencer, J. H., Ramsey theory. Second ed.,Wiley-Interscience Series in Discrete Mathematics and Optimization,Wiley-Interscience, New York, 1990. Google Scholar

[3] [3] Nešetřil, J., Metric spaces are Ramsey. European J. Combin. 28 (2007), no. 1, 457–468. Google Scholar | DOI

[4] [4] Nešetřil, J., Ramsey classes of topological spaces and metric spaces. Ann. Pure Appl. Logic 143 (2006), no. 1–3, 147–154. Google Scholar | DOI

[5] [5] Nešetřil, J. and Rödl, V., Partitions of finite relational and set systems. J. Combinatorial Theory Ser. A 22 (1977), no. 3, 289–312. Google Scholar

[6] [6] Nešetřil, J. and Rödl, V., Ramsey classes of set systems. J. Comb. Theory Ser. A 34 (1983), no. 2, 183–201. Google Scholar | DOI

[7] [7] Schmerl, J., Countable homogeneous partially ordered sets. Algebra Universalis 9 (1979), no. 3,317–321. Google Scholar | DOI

[8] [8] Sokić, M., Ramsey property of posets and related structures. Ph.D. dissertation, University of Toronto, ProQuest LLC, Ann Arbor, MI, 2010. Google Scholar

[9] [9] Sokić, M., Ramsey properties of finite posets. Order 29 (2012), no. 1, 1–30. Google Scholar | DOI

[10] [10] Sokić, M., Ramsey properties of finite posets II. Order 29 (2012), no. 1, 31–47. Google Scholar | DOI

[11] [11] Sokić, M., Ramsey property, ultrametric spaces, finite posets, and universal minimal flows. Israel J. Math. 194 (2013), no. 2, 609–640. Google Scholar | DOI