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We give a classification of all the cycle types occurring in the automorphism groups of countable homogeneous graphs. [ABSTRACT FROM AUTHOR]

We give a classification of certain continuous maps from the unit interval (or any non-trivial closed bounded interval of the real line) to itself up to conjugacy. The maps studied are those for which there is a partition of the interval into finitely many subintervals whose endpoints form a cycle of the map, and on each of which the map is monotonic. [ABSTRACT FROM PUBLISHER]

We discuss the solubility of equations of the form w = g, where w is a word (an element of a free group FX) and g is an element of a given group G. A word for which this equation is soluble for every g ∈ G is said to be universal for G. It is conjectured that a word is universal for the automorphism group of the random graph if and only if it cannot be written as a proper power, corresponding to the results of [Randall Dougherty and Jan Mycielski. Representations of infinite permutations by words (II). Proc. Amer. Math. Soc. 127 (1999), 2233–43.], [Roger C. Lyndon. Words and infinite permutations. In Mots, Lang. Raison Calc. (Hermès, 1990), pp. 143–152.], [Jan Mycielski. Representations of infinite permutations by words. Proc. Amer. Math. Soc. 100 (1987), 237–241.], where the same necessary and sufficient condition was established for infinite symmetric groups. We prove various special cases. A key ingredient is the use of ‘generic’ automorphisms, and elements which suitably approximate them, called ‘special’. [ABSTRACT FROM AUTHOR]