Invariant measures concentrated on countable structures

Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraisse limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.Comment: 46 pages, 2 figures. Small changes following referee suggestion

On The Relational Width of First-Order Expansions of Finitely Bounded Homogeneous Binary Cores with Bounded Strict Width

The relational width of a finite structure, if bounded, is always (1,1) or (2,3). In this paper we study the relational width of first-order expansions of finitely bounded homogeneous binary cores where binary cores are structures with equality and some anti-reflexive binary relations such that for any two different elements a, b in the domain there is exactly one binary relation R with (a, b) in R. Our main result is that first-order expansions of liberal finitely bounded homogeneous binary cores with bounded strict width have relational width (2, MaxBound) where MaxBound is the size of the largest forbidden substructure, but is not less than 3, and liberal stands for structures that do not forbid certain finite structures of small size. This result is built on a new approach and concerns a broad class of structures including reducts of homogeneous digraphs for which the CSP complexity classification has not yet been obtained.Comment: A long version of an extended abstract that appeared in LICS 202

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Reducts of structures and maximal-closed permutation groups

Answering a question of Junker and Ziegler, we construct a countable first order structure which is not ω-categorical, but does not have any proper nontrivial reducts, in either of two senses (model-theoretic, and group-theoretic). We also construct a strongly minimal set which is not ω-categorical but has no proper nontrivial reducts in the model-theoretic sense

Further combinatorial constructions for optimal frequency-hopping sequences

AbstractFrequency-hopping multiple-access (FHMA) spread-spectrum communication systems employing multiple frequency shift keying as data modulation technique were investigated by Fuji-Hara, Miao and Mishima [R. Fuji-Hara, Y. Miao, M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inform. Theory 50 (2004) 2408–2420] from a combinatorial approach, where a correspondence between frequency-hopping (FH) sequences and partition-type cyclic difference packings was established, and several combinatorial constructions were provided for FHMA systems with a single optimal FH sequence. In this paper, by means of this correspondence, we describe more combinatorial constructions for such optimal FH sequences. As a consequence, more new infinite series of optimal FH sequences are obtained