# The extremal number of the subdivisions of the complete bipartite graph

## TL;DR

This paper determines the maximum number of edges in large graphs avoiding subdivisions of complete bipartite graphs, confirming a conjecture and improving existing bounds.

## Contribution

It proves the extremal number for subdivisions of complete bipartite graphs, settling a conjecture and refining previous bounds.

## Key findings

- Established tight bounds for extremal numbers of subdivided bipartite graphs.
- Confirmed the conjecture of Conlon--Janzer--Lee for large t.
- Improved upon previous results by Jiang--Qiu and others.

## Abstract

For a graph $F$, the $k$-subdivision of $F$, denoted $F^k$, is the graph obtained by replacing the edges of $F$ with internally vertex-disjoint paths of length $k$. In this paper, we prove that $\mathrm{ex}(n,K_{s,t}^k)=O(n^{1+\frac{s-1}{sk}})$, which is tight for $t$ sufficiently large. This settles a conjecture of Conlon--Janzer--Lee, and improves on a substantial body of work by Conlon--Janzer--Lee and Jiang--Qiu.