# Induced subgraphs of graphs with large chromatic number. X. Holes of specific residue

## TL;DR

This paper proves that graphs with large chromatic number necessarily contain either large complete subgraphs or induced cycles of all lengths modulo any positive integer, extending previous results and confirming conjectures linking chromatic number and topological properties.

## Contribution

It unifies and extends prior results by showing the existence of specific induced cycles in graphs with large chromatic number for all residues modulo any positive integer.

## Key findings

- Graphs with large chromatic number contain induced cycles of all lengths modulo k.
- Confirmed conjectures relating chromatic number to the homology of independence complexes.
- Established a comprehensive framework for induced cycles in high chromatic graphs.

## Abstract

A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of previous papers, showing that, for every positive integer k, every graph with large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k. As an application, we prove two conjectures of Kalai and Meshulam from the 1990's connecting the chromatic number of a graph with the homology of its independence complex.