Why concept lattices are large: extremal theory for generators, concepts, and VC-dimension

Abstract

A unique type of subcontexts is always present in formal contexts with many concepts: the contranominal scales. We make this precise by giving an upper bound for the number of minimal generators (and thereby for the number of concepts) of contexts without contranominal scales larger than a given size. We give an interpretation of this bound in terms of the Vapnik–Chervonenkis dimension of the concept lattice. Extremal contexts are constructed which meet this bound exactly. They are completely classified.

Keywords:

• Number of concepts
• minimal generators
• upper bound
• Vapnik–Chervonenkis dimension
• shattered sets
• contranominal scales

Acknowledgements

We would like to deeply thank Bernhard Ganter for the invaluable feedback and fruitful discussions.

Notes

No potential conflict of interest was reported by the authors.