Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- On maximum size anti-Pasch sets of triples
- Some simple 7–designs
- Inscribed bundles, Veronese surfaces and caps
- Embedding partial geometries in Steiner designs
- Finite geometry after Aschbacher's Theorem: PGL(n, q) from a Kleinian viewpoint
- The Hermitian function field arising from a cyclic arc in a Galois plane
- Intercalates everywhere
- Difference sets: an update
- Computational results for the known biplanes of order 9
- A survey of small embeddings for partial cycle systems
- Rosa triple systems
- Searching for spreads and packings
- A note on Buekenhout-Metz unitals
- Elation generalized quadrangles of order (q2, q)
- Uniform parallelisms of PG(3, 3)
- Double-fives and partial spreads in PG(5, 2)
- Rank three geometries with simplicial residues
- Generalized quadrangles and the Axiom of Veblen
- Talks
- Participants
Intercalates everywhere
Published online by Cambridge University Press: 04 November 2009
- Frontmatter
- Contents
- Preface
- Introduction
- On maximum size anti-Pasch sets of triples
- Some simple 7–designs
- Inscribed bundles, Veronese surfaces and caps
- Embedding partial geometries in Steiner designs
- Finite geometry after Aschbacher's Theorem: PGL(n, q) from a Kleinian viewpoint
- The Hermitian function field arising from a cyclic arc in a Galois plane
- Intercalates everywhere
- Difference sets: an update
- Computational results for the known biplanes of order 9
- A survey of small embeddings for partial cycle systems
- Rosa triple systems
- Searching for spreads and packings
- A note on Buekenhout-Metz unitals
- Elation generalized quadrangles of order (q2, q)
- Uniform parallelisms of PG(3, 3)
- Double-fives and partial spreads in PG(5, 2)
- Rank three geometries with simplicial residues
- Generalized quadrangles and the Axiom of Veblen
- Talks
- Participants
Summary
Abstract
The counting of the number of Intercalates, 2 × 2 subsquares, possible in a latin square of side n is in general a hard problem. N2–Free latin squares, those for which there are no intercalates, are known to exist for n ≠ 1,2,4. N2–complete latin squares, those which have the property that they have the maximum number of N2's possible, must be isotopic to and thus of side 2k. The maximum for n ≠ 2k is in general unknown. We propose an intermediate possibility, that of N2–ubiquitous. A latin square is N2 ubiquitous if and only if every cell aij is contained in some 2 × 2 subsquare. We show these exist for n ≠ 1,3,5,7. It is also determined for which n, C–ubiquitous latin squares exist for every partial latin square, C with four cells. We also enumerate the number of times each 4-cell configuration can appear in a latin square and show that this number depends only on n and the number of intercalates.
- Type
- Chapter
- Information
- Geometry, Combinatorial Designs and Related Structures , pp. 69 - 88Publisher: Cambridge University PressPrint publication year: 1997