# Rich subcontexts

## TL;DR

This paper introduces a local operation on finite binary relations that preserves or increases the number of Galois-closed sets and can precisely control the VC-dimension and the count of closed sets.

## Contribution

It demonstrates the existence of a specific subrelation that maximizes Galois-closed sets for given VC-dimension and ground set sizes.

## Key findings

- Existence of subrelations with at least half the Galois-closed sets.
- Construction of relations with specified VC-dimension and maximum Galois-closed sets.
- Theoretical bounds on the number of Galois-closed sets for given parameters.

## Abstract

For a finite binary relation, we show a local operation which does not decrease its number of (Galois-)closed sets and eventually increases its (Vapnik-Chervonenkis)-dimension. Specifically, we show that there always exist a pair of elements, one belonging to each ground set, such that the subrelation not relating any of those elements has at least half of the Galois-closed sets. As a consequence, for each triple (n,m,k) there exists a binary relation with VC-dimension precisely k and maximum number of Galois-closed sets, such maximum being over all binary relations having ground sets with precisely n and m elements.