# A Kruskal-Katona type result and applications

## TL;DR

This paper generalizes the Kruskal-Katona theorem by replacing shadows with images under increasing functions, showing compressed sets as solutions and exploring applications to simplicial complexes.

## Contribution

It introduces a new minimization problem inspired by Kruskal-Katona, with compressed sets identified as optimal solutions and applications to simplicial complexes.

## Key findings

- Compressed sets solve the new minimization problem.
- The approach extends Kruskal-Katona to monoid actions.
- Applications to simplicial complexes are demonstrated.

## Abstract

Inspired by the Kruskal-Katona theorem a minimization problem is studied, where the role of the shadow is replaced by the image of the action of the monoid of increasing functions. One of our main results shows that compressed sets are a solution to this problem. Several applications to simplicial complexes are discussed.