Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Ramsey number of Steiner systems

Ayush Basu, Daniel Dobak, Vojtěch Rödl, Marcelo Sales

Abstract

A $k$-uniform hypergraph $H$ is called a partial $(k,\ell)$-system if every set of $\ell$ vertices of $V(H)$ is contained in at most one edge of $H$. We prove the existence of a partial $(k,k-1)$-system $H$ whose Ramsey number with $r \geq 4$ colors grows as a tower of height $k-1$.

On Ramsey number of Steiner systems

Abstract

A -uniform hypergraph is called a partial -system if every set of vertices of is contained in at most one edge of . We prove the existence of a partial -system whose Ramsey number with colors grows as a tower of height .
Paper Structure (4 sections, 2 theorems, 6 equations, 2 figures)

This paper contains 4 sections, 2 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Stepping up lemma
  3. Preliminaries
  4. Binary trees, ancestors and descendants

Key Result

Theorem 1.1

For every $k\geq 3$, there exists a positive constant $c_k$ and an integer $h_0:=h_0(k)$ such that the following holds. For every integer $h\geq h_0$, there exists a $(k,k-1)$-system $H$ on $h$ vertices such that

Figures (2)

  • Figure 1: A member of $\mathcal{F}^{(3)}_I(n)$ with $I=\{1,2\}$, vertex set $\{x_0\}\cup\left\{x_{ij}:\;ij\in [n]^{( 2)}\right\}\cup\{x_1,\dots,x_n\}$ and edge set $F = \{x_{0},x_1,x_2\}\cup\left\{ \{x_{ij}, x_i, x_j\}: ij\in [n]^{(2)} \right\}$.
  • Figure 2: In this example where $N = 2$, the leaves 1 and 2 are left descendants of $u$ and $a(1,2)$ is the left child of $u$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Definition 2.1: The families $\mathcal{F}_I^{(k)}(n)$, $\mathop{\mathrm{rev}}\nolimits\mathcal{F}_I^{(k)}(n)$ and $\mathcal{F}^{*}_I(n,k)$
  • Remark 2.2
  • Theorem 2.3