Descriptive Complexity and Reflective Properties of Combinatorial Configurations

Abstract

We study the Ramsey theoretic properties of combinatorial configurations which are generated by infinite binary strings which are random in the sense of Kolmogorov-Chaitin. Introduction In this paper we investigate the properties of combinatorial configurations which are generated or encoded by infinite binary strings which are random in the sense of Kolmogorov-Chaitin [3,9,4]. Using ideas from Ramsey theory, we introduce various means of assessing the degree of reflectiveness (self-similarity) of classes of combinatorial configurations and show how Kolmogorov-Chaitin strings (or KC strings) bring these reflective properties into realisation. We show for many categories C of combinatorial configurations (including graphs, posets, lattices and trees) that one can find a countable object U with the following properties. The object U is universal in the sense that every finite object in C can be embedded into U. Moreover, if /? is any finite object in C, we can use an arbitrary KC string e (i) for serving as a 2-colouring of the copies (images under embeddings) of fi in U and (ii) for generating a copy U' of U in U such that all the copies of fi which are contained in U' are of the same colour with respect to the colouring e. Since an infinite binary string is, with probability 1, a KC string, one can therefore conclude that for a random and uniform 2-colouring x of the copies of ft in U there is, with probability 1, a copy U' of U in U such that all the copies of fi in U' are monochromatic with respect to xIn a sense, KC strings are themselves highly reflective objects. Indeed, an important factor for the wjde acceptance of Kolmogorov's definition [9] of random strings has been that strings which are random according to his definition also satisfy the Church-Von Mises-Loveland criteria for randomness (see [18] for a historical survey). In particular, the sequence of outcomes brought about by a recursive gambling strategy against a KC string will itself be a KC string. Another way of depicting the reflective properties of a KC string e: N -*• {0,1} is by considering its graph-theoretic representation. Let Km be the complete graph on the set of natural numbers and let ex,e2, ...be a recursive enumeration without repetition of the edges of K^. Let G = (N, Ve) be the subgraph of Kx such that, for every ie N, we have ete Ve if and only if e(i) = 1. It is shown in Section 4 that Ge is isomorphic to the random graph of Rado [16]. In particular, Ge has every countable graph as an induced subgraph. Received 9 December 1992. 1991 Mathematics Subject Classification 68Q30. J. London Math. Soc. (2) 54 (1996) 199-208