# Extremal graphs for edge blow-up of graphs

## TL;DR

This paper determines the maximum number of edges in large graphs that avoid containing the edge blow-up of any given graph, extending classical Turán theory to more complex graph constructions.

## Contribution

It generalizes Turán number calculations to the edge blow-up of arbitrary graphs, providing a comprehensive solution to a previously studied special case.

## Key findings

- Derived explicit Turán numbers for edge blow-up of general graphs.
- Extended classical extremal graph theory to complex graph blow-ups.
- Solved a longstanding problem in extremal combinatorics.

## Abstract

Given a graph $H$ and an integer $p$, the {\it edge blow-up} of $H$, denoted as $H^{p+1}$, is the graph obtained from replacing each edge in $H$ by a clique of size $p+1$ where the new vertices of the cliques are all different. The Tur\'{a}n numbers for edge blow-up of matchings were first studied by Erd\H{o}s and Moon. In this paper, we determine the Tur\'{a}n numbers for edge blow-up of general graphs.