Generalized Tur\'an problems for $K_{2,t}$

TL;DR

This paper investigates the generalized Turán function involving the bipartite graph $K_{2,t}$, providing order of magnitude results for trees and asymptotics for various cases, advancing extremal graph theory understanding.

Contribution

It determines the order of magnitude of $ex(n,H,K_{2,t})$ for trees and establishes asymptotics for a broad class of trees and most cases of $ex(n,K_{2,t},H)$, extending Turán problem knowledge.

Findings

Order of magnitude of $ex(n,H,K_{2,t})$ for trees is established.

Asymptotics for $ex(n,K_{2,t},H)$ are determined in most cases.

Results advance understanding of Turán problems involving bipartite graphs.

Abstract

We study the generalized Tur\'an function $ex(n,H,F)$, when $H$ or $F$ is $K_{2,t}$. We determine the order of magnitude of $ex(n,H,K_{2,t})$ when $H$ is a tree, and determine its asymptotics for a large class of trees. We also determine the asymptotics of $ex(n,K_{2,t},H)$ in most cases.