We introduce and study relations on words which generalize the factor relation, being restrictions of the subword relation. We give an equivalent condition for the nite basis property for these relations which generalizes the well-known theorem of Higman. Some language-theoretic gaps for in nite antichains are also presented.

This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters). Mathematics Subject Classification (2000). Primary 20F65; Secondary 20F10.

We construct the graphs of Hoffman-Singleton, Higman-Sims and McLaughlin from certain relations on the set of non-singular conics totally tangent to the Hermitian curve of degree 6 in characteristic 5. We then interpret this geometric construction in terms of the subgroup structure of the automorphism group of this Hermitian curve.

We prove that groups acting boundedly and order-primitively on linear orders or acting extremly proximality on a Cantor set (the class including various Higman-Thomson groups and Neretin groups of almost automorphisms of regular trees, also called groups of spheromorphisms) are uniformly simple. Explicit bounds are provided.

Let P be a point set with n elements in general position. A triangulation T of P is a set of triangles with disjoint interiors such that their union is the convex hull of P , no triangle contains an element of P in in its interior, and the vertices of the triangles of T are points of P . Given T we define a graph G(T ) whose vertices are the triangles of T , two of which are adjacent if they sh...

Given two strings x, y ∈ Σ∗, say that x is a subsequence of y (denoted x y) if x results from removing zero or more characters from y. For a language L ⊆ Σ∗, define SUBSEQ(L) to be the set of all subsequences of strings in L. We give a new proof of a result of Higman, which states, If L is any language over a finite alphabet, then SUBSEQ(L) is regular. Higman’s original proof makes use of the t...

A review is given of some of the mathematical research of Bruce Rothschild, emphasizing his results in combinatorial theory, especially that part known as Ramsey Theory. Special emphasis is given to the Graham-Rothschild Parameter Sets Theorem, its consequences, and some extensions.