Induced subdivisions of $K_{d+1}$ in graphs of high girth

TL;DR

This paper proves that graphs with very high minimum degree and girth necessarily contain large induced subdivisions of complete graphs, addressing a longstanding open problem in graph theory.

Contribution

It establishes that for sufficiently large parameters, high girth and degree graphs contain induced subdivisions of large complete graphs, solving a problem posed by K"uhn and Osthus.

Findings

Graphs with minimum degree and girth at least 10^8 contain induced subdivisions of K_{k+1}.

The result confirms the existence of complex substructures in sparse, high-girth graphs.

Addresses a problem posed by K"uhn and Osthus regarding induced subdivisions in graphs.

Abstract

In this paper, we show that for all $k\geq 10^8$, every graph with minimum degree $k$ and girth at least $10^8$ contains an induced subdivision of a $K_{k+1}$. This answers a problem asked by K\"uhn and Osthus (originally attributed to Shi).