On a Problem in Combinations

A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

http://emis.matem.unam.mx/journals/EJC/Surveys/ds16.pdf

Dynamic Cage Survey

Ancillary Mathematics By Prof. H. S. W. Massey and H. Kestelman. Pp. xvi + 990. (London: Sir Isaac Pitman and Sons, Ltd., 1959.) 75s. net.

/pdf/mathematics-for-the-physical-sciences-5gfvv151jq.pdf

Mathematics for the Physical Sciences

Finite linear spaces with flag-transitive groups

This is a report of research carried out during 1992 and 1993 in which three different automated reasoning programs—DDPP, FINDER, and MGTP—were applied to a series of exhaustive search problems in the theory of quasigroups. All three of the programs succeeded in solving previously open problems concerning the existence of quasigroups satisfying certain additional conditions. Using different programs has allowed us to cross-check the results, helping reliability. We find this research interesting from several points of view: first, it brings techniques from the field of automated reasoning to bear on a rather different problem domain from that which motivated their development; second, investigating such hard problems leads us to push the limits of what our systems have achieved; and finally, it involves us in serious philosophical issues concerning essentially computational proofs.

Automated reasoning and exhaustive search: Quasigroup existence problems☆

On Buekenhout-Metz unitals of odd order

An elliptic semiplane

Maximal cliques in the Paley graph of square order

We define a nest of reguli to be a collection P of reguli in a regular spread S of PG(3, q) such that every line of S is contained in exactly 0 or 2 reguli of P. Let U denote the lines of S contained in the reguli of some nest. If V is a partial spread of PG[3,q) covering the same points as U but having no lines in common with U , then V will be called a replacement set for U. Clearly, (S—U) U V is a spread of PG(3, q) , yielding a (potentially new) translation plane of order q 2 which is 2–dimensional over its kernel. Nests of size (q+3)/2 were first studied (under another name) by Bruen and later by many others. Whether such (q+3)/2- nests exist for q 13 and whether such nests are necessarily reversible are still open questions. In this paper we consider nests of size q. We exhibit an infinite family of ?-nests, one for each odd prime q , and show that each nest is reversible. The translation planes so obtained appear to be new, at least for q ≥ 11.

R. D. Baker

Papers

On Buekenhout-Metz unitals of odd order

An elliptic semiplane

Maximal cliques in the Paley graph of square order

A New Class of Translation Planes

Mixed partitions of PG(5, q )