Tur\'an Density of $2$-edge-colored Bipartite Graphs with Application on $\{2, 3\}$-Hypergraphs

TL;DR

This paper determines the Turán densities for all 2-edge-colored bipartite graphs and applies these findings to Turán problems in hypergraphs with edges of size 2 and 3.

Contribution

It provides a complete characterization of Turán densities for 2-edge-colored bipartite graphs and demonstrates their application to hypergraph Turán problems.

Findings

Turán densities for all 2-edge-colored bipartite graphs are explicitly determined.

The results are applied to solve Turán problems in {2,3}-hypergraphs.

New methods are introduced for analyzing edge-colored graph Turán problems.

Abstract

We consider the Tur\'an problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$ and $E_b$ do not have to be disjoint. The Tur\'an density $\pi(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/{n\choose 2}$ is the formula to measure the edge density of $G_n$. We will determine the Tur\'an densities of all $2$-edge-colored bipartite graphs. We also give an important application of our study on the Tur\'an problems of $\{2, 3\}$-hypergraphs.