Improved bounds for the extremal number of subdivisions

TL;DR

This paper improves the upper bounds on the extremal number of subdivisions of complete graphs, refining previous results and providing tighter asymptotic estimates for large graphs.

Contribution

It establishes a new, sharper upper bound for the extremal number of subdivisions of complete graphs, advancing understanding of their extremal properties.

Findings

New upper bound: ex(n, H_t) ≤ C' n^{3/2 - 1/(4t-6)}

Improved asymptotic estimates for subdivisions of K_t

Refinement over previous bounds by Conlon and Lee

Abstract

Let $H_t$ be the subdivision of $K_t$. Very recently, Conlon and Lee have proved that for any integer $t\geq 3$, there exists a constant $C$ such that $\text{ex}(n,H_t)\leq Cn^{3/2-1/6^t}$. In this paper, we prove that there exists a constant $C'$ such that $\text{ex}(n,H_t)\leq C'n^{3/2-\frac{1}{4t-6}}$.