# The extremal number of longer subdivisions

## TL;DR

This paper proves conjectures regarding the extremal number of longer subdivisions of multigraphs, establishing bounds that depend on the subdivision length and confirming the predicted growth rates.

## Contribution

It confirms Conlon and Lee's conjectures by proving the extremal number bounds for longer subdivisions of multigraphs and simple graphs.

## Key findings

- Proved the extremal number bounds for the $(k-1)$-subdivision of multigraphs.
- Confirmed the conjecture for simple graphs with a positive epsilon gap.
- Established that the extremal number grows as $O(n^{1+1/k})$ for even $k$.

## Abstract

For a multigraph $F$, the $k$-subdivision of $F$ is the graph obtained by replacing the edges of $F$ with pairwise internally vertex-disjoint paths of length $k+1$. Conlon and Lee conjectured that if $k$ is even, then the $(k-1)$-subdivision of any multigraph has extremal number $O(n^{1+\frac{1}{k}})$, and moreover, that for any simple graph $F$ there exists $\varepsilon>0$ such that the $(k-1)$-subdivision of $F$ has extremal number $O(n^{1+\frac{1}{k}-\varepsilon})$. In this paper, we prove both conjectures.