论文信息 - Shift graphs on precompact families of finite sets of natural numbers

Shift graphs on precompact families of finite sets of natural numbers

Abstract We study graphs defined on families of finite sets of natural numbers and their chromatic properties. Of particular interest are graphs for which the edge relation is given by the shift. We show that when considering shift graphs with infinite chromatic number, one can center attention on graphs defined on precompact thin families. We define a quasi-order relation on the collection of uniform families defined in terms of homomorphisms between their corresponding shift graphs, and show that there are descending ω 1 -sequences. Specker graphs are also considered and their relation with shift graphs is established. We characterize the family of Specker graphs which contain a homomorphic image of a shift graph.

Stevo Todorcevic | Carlos A. Di Prisco | S. Todorcevic | C. D. Prisco

[1] Stevo Todorcevic,et al. Canonical forms of shift-invariant maps on [N]infinity , 2006, Discret. Math..

[2] R. Rado,et al. A Construction of Graphs without Triangles having Pre‐Assigned Order and Chromatic Number , 1960 .

[3] Vojtech Rödl,et al. Shift graphs and lower bounds on Ramsey numbers rk(l; r) , 1995, Discret. Math..

[4] Stevo Todorcevic,et al. BOREL CHROMATIC NUMBERS , 1999 .

[5] Stevo Todorcevic,et al. Ramsey Methods in Analysis , 2005 .

[7] Stevo Todorcevic,et al. Oscillations of Sets of Integers , 1998 .

[8] Finite order spreading models , 2011, 1105.2732.

[9] M. Behzad,et al. Chromatic numbers of infinite graphs , 1976, J. Comb. Theory, Ser. B.

[10] Roger C. Entringer,et al. Arc Colorings of Digraphs , 1972 .

[11] Vojtech Rödl,et al. Partition theorems for systems of finite subsets of integers , 1982, Discret. Math..

[12] J. Lopez-Abad,et al. Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces , 2005, math/0510407.

[13] N. H. Williams,et al. Combinatorial Set Theory , 1977 .

[14] C. St. J. A. Nash-Williams,et al. On well-quasi-ordering transfinite sequences , 1965, Mathematical Proceedings of the Cambridge Philosophical Society.

[15] Ernst Specker,et al. Teilmengen von Mengen mit Relationen , 1956 .

[16] P. Erdos,et al. On chromatic number of graphs and set-systems , 1966 .