# On vertex-induced weighted Tur\'an problems

## TL;DR

This paper investigates the vertex-induced weighted Turán problem, characterizes extremal structures for $K_{l}$-free graphs under sum-edge-weight functions, and generalizes the Erdős-Stone theorem for weighted graphs.

## Contribution

It provides a characterization of extremal $K_{l}$-free graphs with sum-edge-weight functions and extends the Erdős-Stone theorem to weighted graphs with vertex-induced weights.

## Key findings

- Characterization of extremal $K_{l}$-free graphs under sum-edge-weight functions
- Generalized Erdős-Stone theorem for weighted graphs with vertex-induced weights
- New insights into weighted Turán problems

## Abstract

Recently, Bennett et al. introduced the vertex-induced weighted Tur\'an problem. In this paper, we consider their open Tur\'an problem under sum-edge-weight function and characterize the extremal structure of $K_{l}$-free graphs. Based on these results, we propose a generalized version of the Erd\H{o}s-Stone theorem for weighted graphs under two types of vertex-induced weight functions.