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On the Ramsey numbers of fans and stars

Louis DeBiasio, Tucker Wimbish

Abstract

Let $F_n$ be the graph on $2n+1$ vertices consisting of $n$ triangles meeting at a single vertex. After a number of improvements over the years, it is currently known that the Ramsey number of $F_n$ is between $4.5n-5$ (Chen, Yu, Zhao) and $(5+\frac{1}{6})n+O(1)$ (Dvo{ř}{á}k and Metrebian). We improve both of these bounds as follows $$4.732n\approx (3+\sqrt{3})n-8< R(F_n)\leq (5+o(1))n.$$ Additionally, as it relates to the lower bound on $R(F_n)$ (and for which nothing was known when $n< m< n(n-1)$), we determine the Ramsey numbers of stars vs.~fans, within a constant, as follows $$R(K_{1,m}, F_{n})= \begin{cases} m+2n-\frac{1+(-1)^{m}}{2}, & m\leq n \frac{3m+\sqrt{m^2+8n^2}}{2}+Θ(1), & m>n \end{cases}. $$ In particular, we have $R(K_{1,2n}, F_n)=(3+\sqrt{3})n+Θ(1)$.

On the Ramsey numbers of fans and stars

Abstract

Let be the graph on vertices consisting of triangles meeting at a single vertex. After a number of improvements over the years, it is currently known that the Ramsey number of is between (Chen, Yu, Zhao) and (Dvo{ř}{á}k and Metrebian). We improve both of these bounds as follows Additionally, as it relates to the lower bound on (and for which nothing was known when ), we determine the Ramsey numbers of stars vs.~fans, within a constant, as follows In particular, we have .

Paper Structure

This paper contains 14 sections, 16 theorems, 102 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Ramsey
  3. Dirac
  4. Turán
  5. Overview
  6. Notation
  7. Stars vs. fans
  8. Lower bound
  9. Upper bound
  10. Upper bounds on the Ramsey numbers of fans
  11. Lemmas for fans
  12. Main proof
  13. Conclusion and open problems
  14. Appendix: A lower bound on the Ramsey number of $K_{1,2n}$ vs. $F_n$

Key Result

Theorem 1.1

For all $\epsilon>0$ and all $n\geq \frac{384}{\epsilon^2}$,

Figures (5)

  • Figure 1: A graph on $N$ vertices with minimum degree at least $N-m$ having no copy of $F_n$. The values of $a,b,\sigma$ are as in the proof.
  • Figure 2: Counting the edges between $X_1\cup X_2$ and $Y=V(G)\setminus (X_1\cup X_2)$
  • Figure 3: Fan extension lemma
  • Figure 4:
  • Figure 5: A graph on $N$ vertices with minimum degree at least $N-2n$ having no copy of $F_n$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Theorem 2.1: Gale Gale, Ryser Ry
  • Lemma 2.2
  • proof
  • Example 2.3: General example
  • proof
  • Remark 2.4
  • Theorem 2.5: Edmonds Ed, Gallai Gal
  • ...and 31 more