Transversals in Linear Uniform Hypergraphs

This book gives the state-of-the-art on transversals in linear uniform hypergraphs. The notion of transversal is fundamental to hypergraph theory and has been studied extensively. Very few articles have discussed bounds on the transversal number for linear hypergraphs, even though these bounds are integral components in many applications. This book is one of the first to give strong non-trivial bounds on the transversal number for linear hypergraphs. The discussion may lead to further study of those problems which have not been solved completely, and may also inspire the readers to raise new questions and research directions. The book is written with two readerships in mind. The first is the graduate student who may wish to work on open problems in the area or is interested in exploring the field of transversals in hypergraphs. This exposition will go far to familiarize the student with the subject, the research techniques, and the major accomplishments in the field. The photographs included allow the reader to associate faces with several researchers who made important discoveries and contributions to the subject. The second audience is the established researcher in hypergraph theory who will benefit from having easy access to known results and latest developments in the field of transversals in linear hypergraphs.

1. Introduction.- 2. Linear Intersection Hypergraphs.- 3. Finite Affine Planes and Projective Planes.- 4 . The Tuza Constants.- 5. The Tuza Constant c4.- 6. The Tuza Constant ck for k Large.- 7. The West Bound.- 8. The Deficiency of a Hypergraph.- 9. The Tuza Constant q4.- 10. The Tuza Constant qk for Large k.- 11. The Cap Set Problem.- 12. Partial Steiner Triple Systems.- 13. Upper Transversals in Linear Hypergraphs.- 14. Strong Tranversals in Linear Hypergraphs.- 15. Conjectures and Open Problems.- References.- Glossary.