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It is shown that the maximum size of a set $S$ of vectors of a $k$ -dimensional vector space over $𝔽_{q}$ , with the property that every subset of size $k$ is a basis, is at most $q + 1$ , if $k \leq p$ , and at most $q + k - p$ , if $q \geq k \geq p + 1 \geq 4$ , where $q = p^{k}$ and $p$ is prime. Moreover, for $k \leq p$ , the sets $S$ of maximum size are classified, generalising Beniamino Segre’s “arc is a conic” theorem. These results have various implications. One such implication is that a $k \times (p + 2)$ matrix, with $k \leq p$ and entries from $𝔽_{p}$ , has $k$ columns which are linearly dependent. Another is...

On Squashed Designs.

M. Deza, Frankl P. (1986)

Discrete & computational geometry

On Subgraphs as Matroid Cells.

J.M.S. Simoes Pereira (1972)

Mathematische Zeitschrift

On sums over partially ordered sets.

Dohmen, Klaus (1999)

The Electronic Journal of Combinatorics [electronic only]

On the center of the fundamental group of the complement of a hyperplane arrangement.

Cordovil, Raul (1994)

Portugaliae Mathematica

On the Coordinatization of Oriented Matroids.

J. Bokowski, Bernd Sturmfels (1986)

Discrete & computational geometry

On the excluded minors for matroids of branch-width three.

Hliněný, Petr (2002)

The Electronic Journal of Combinatorics [electronic only]

On the local linear independence of translates of a box spline

Wolfgang Dahmen, Charles Micchelli (1985)

Studia Mathematica

On the maximum positive semi-definite nullity and the cycle matroid of graphs.

Van Der Holst, Hein (2009)

ELA. The Electronic Journal of Linear Algebra [electronic only]

On the notion of transversal of a sum of matroids

Cordovil, Raúl, Dias da Silva, J.A., Fonseca, Amélia (1988)

Portugaliae mathematica

On the number of matroids on a finite set.

Dukes, W.M.B. (2004)

Séminaire Lotharingien de Combinatoire [electronic only]

On the rank of random subsets of finite affine geometry

Wojciech Kordecki (2000)

Discussiones Mathematicae Graph Theory

The aim of the paper is to give an effective formula for the calculation of the probability that a random subset of an affine geometry AG(r-1,q) has rank r. Tables for the probabilities are given for small ranks. The expected time to the first moment at which a random subset of an affine geometry achieves the rank r is derived.

On the Realization of Convex Polytopes, Euler's Formula and Möbius Functions (Short Communication)

BERNT LINDSTRÖM (1971)

Aequationes mathematicae

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