[Paper Review] Size-Ramsey numbers of powers of hypergraph trees and long subdivisions

Shoham Letzter, Alexey Pokrovskiy+1 more|arXiv (Cornell University)|Mar 2, 2021

Limits and Structures in Graph Theory30 references4 citations

TL;DR

This paper establishes that the $s$-colour size-Ramsey number of the $t$-power of an $r$-uniform tight path on $n$ vertices is linear in $n$, for all fixed $r, s, t$. It further proves that powers of bounded-degree hypergraph trees and long subdivisions of bounded-degree hypergraphs also have linear size-Ramsey numbers, resolving a conjecture by Dudek, La Fleur, Mubayi, and Rödl on the linearity of size-Ramsey numbers for hypergraph paths.

ABSTRACT

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers that are linear in the number of vertices. This extends and strongly generalises recent results about the linearity of size-Ramsey numbers of powers of bounded degree trees and of long subdivisions of bounded degree graphs.

Motivation & Objective

• To resolve a conjecture by Dudek, La Fleur, Mubayi, and Rödl on the linearity of size-Ramsey numbers for $r$-uniform tight paths.
• To generalize Beck's result on path size-Ramsey numbers to hypergraphs and their powers.
• To establish linear size-Ramsey numbers for powers of bounded-degree hypergraph trees and long subdivisions of bounded-degree hypergraphs.
• To extend recent results on powers of bounded-degree graphs to the hypergraph setting.

Proposed method

• Defining the $t$-power of an $r$-uniform tight path as the hypergraph whose edges are $r$-sets contained in intervals of length $r + t - 1$.
• Using a probabilistic construction to build a host hypergraph $G$ with $O(n)$ edges that guarantees a monochromatic copy of the $t$-power of a tight path in any $s$-edge-colouring.
• Applying a sparse regularity method and embedding techniques tailored for hypergraphs to ensure monochromatic containment.
• Generalizing the argument to hypergraph trees and long subdivisions via structural decomposition and induction on edge connectivity.
• Leveraging known results on bounded-degree graphs and extending them to hypergraphs using hypergraph regularity and density arguments.
• Employing a refined analysis of distance and connectivity in hypergraph trees to control the growth of the size-Ramsey number.

Experimental results

Research questions

• RQ1 Is the $s$-colour size-Ramsey number of the $t$-power of an $r$-uniform tight path linear in $n$ for fixed $r, s, t$?
• RQ2 Do powers of bounded-degree hypergraph trees have linear size-Ramsey numbers?
• RQ3 Do long subdivisions of bounded-degree hypergraphs have linear size-Ramsey numbers?
• RQ4 Can the linearity of size-Ramsey numbers be extended from graphs to hypergraphs beyond tight paths?

Key findings

• The $s$-colour size-Ramsey number of the $t$-power of an $r$-uniform tight path on $n$ vertices is $O(n)$ for all fixed $r, s, t$.
• The size-Ramsey number of powers of bounded-degree hypergraph trees is linear in the number of vertices.
• The size-Ramsey number of long subdivisions of bounded-degree hypergraphs is linear in the number of vertices.
• The result strengthens previous bounds on size-Ramsey numbers for hypergraph paths, resolving the open problem of whether $\hat{r}(P_n^{(r)}) = O(n)$ for $r \geq 4$.
• The paper provides the first generalization of linear size-Ramsey numbers from graphs to hypergraphs beyond simple paths.
• The proof technique extends to hypergraph trees and subdivisions, demonstrating robustness across broader hypergraph families.