Directed graphs and boron trees

Let L 1 and L 2 be two disjoint relational signatures. Let K 1 and K 2 be Ramsey classes of rigid relational structures in L 1 and L 2 respectively. Let K 1 * K 2 be the class of structures in L 1 ? L 2 whose reducts to L 1 and L 2 belong to K 1 and K 2 respectively. We give a condition on K 1 and K 2 which implies that K 1 * K 2 is a Ramsey class. This is an extension of a result of M. Bodirsky.In the second part of this paper we consider classes OS ( 2 ) , OS ( 3 ) , OB and OH which are obtained by expanding the class of finite dense local orders, the class of finite circular directed graphs, the class of finite boron tree structures, and the class of rooted trees respectively with linear orderings. We calculate Ramsey degrees for objects in these classes.

[2]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[3]  Manuel Bodirsky New Ramsey Classes from Old , 2014, Electron. J. Comb..

[4]  Jakub Jasinski Ramsey degrees of boron tree structures , 2013, Comb..

[5]  Peter M. Neumann,et al.  Relations related to betweenness : their structure and automorphisms , 1998 .

[6]  Miodrag Sokic Ramsey Property of Posets and Related Structures , 2011 .

[7]  G. Cherlin,et al.  The Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous N-Tournaments , 1998 .

[8]  M. Sokic,et al.  Ramsey property, ultrametric spaces, finite posets, and universal minimal flows , 2013 .

[9]  Miodrag Sokic,et al.  Ramsey Properties of Finite Posets II , 2012, Order.

[10]  Roland Fraïssé Theory of relations , 1986 .

[11]  Jaroslav Nesetril,et al.  Metric spaces are Ramsey , 2007, Eur. J. Comb..

[12]  Leo Harrington,et al.  Models Without Indiscernibles , 1978, J. Symb. Log..

[13]  P. Cameron,et al.  Oligomorphic permutation groups , 1990 .

[14]  V. Pestov,et al.  Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups , 2003 .

[15]  Keith R. Milliken,et al.  A Ramsey Theorem for Trees , 1979, J. Comb. Theory, Ser. A.

[16]  J. Spencer Ramsey Theory , 1990 .

[17]  Vojtech Rödl,et al.  Ramsey Classes of Set Systems , 1983, J. Comb. Theory, Ser. A.

[18]  Nguyen Van The,et al.  Structural Ramsey theory of metric spaces and topological dynamics of isometry groups , 2008 .

[19]  Vojtech Rödl,et al.  Partitions of Finite Relational and Set Systems , 1977, J. Comb. Theory A.

[20]  Kenneth McAloon,et al.  On Gödel incompleteness and finite combinatorics , 1987, Ann. Pure Appl. Log..

[21]  Peter J. Cameron,et al.  SOME TREELIKE OBJECTS , 1987 .