Exact results for some extremal problems on expansions I

TL;DR

This paper establishes tight bounds and new extremal results for 3-graphs related to expansions of trees, rainbow colorings, and shadow containment, extending and sharpening previous combinatorial theorems.

Contribution

It provides new extremal bounds for 3-graphs avoiding certain expansions, extends anti-Ramsey results, and addresses generalized Turán problems involving shadows.

Findings

Tight upper bounds for 3-graphs avoiding certain tree expansions.

Extended anti-Ramsey results for rainbow colorings of expansions.

Answered a generalized Turán problem related to shadows of expansions.

Abstract

The expansion of a graph $F$, denoted by $F^3$, is the $3$-graph obtained from $F$ by adding a new vertex to each edge such that different edges receive different vertices. For large $n$, we establish tight upper bounds for: The maximum number of edges in an $n$-vertex $3$-graph that does not contain $T^3$ for certain class $\mathcal{T}$ of trees, sharpening (partially) a result of Kostochka--Mubayi--Verstra\"{e}te. The minimum number of colors needed to color the complete $n$-vertex $3$-graph to ensure the existence of a rainbow copy of $F^3$ when $F$ is a graph obtained from some tree $T\in \mathcal{T}$ by adding a new edge, extending anti-Ramsey results on $P_{2t}^3$ by Gu--Li--Shi and $C_{2t}^3$ by Tang--Li--Yan. The maximum number of edges in an $n$-vertex $3$-graph whose shadow does not contain the shadow of $C_{k}^3$ or $T^3$ for $T\in \mathcal{T}$, answering a question of Lv \etal on generalized Tur\'{a}n problems.