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The asymptotics of $r(4,t)$

Sam Mattheus, Jacques Verstraete

Abstract

For integers $s,t \geq 2$, the Ramsey numbers $r(s,t)$ denote the minimum $N$ such that every $N$-vertex graph contains either a clique of order $s$ or an independent set of order $t$. In this paper we prove \[ r(4,t) = Ω\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty\] which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and solves a conjecture of Erdős.

The asymptotics of $r(4,t)$

Abstract

For integers , the Ramsey numbers denote the minimum such that every -vertex graph contains either a clique of order or an independent set of order . In this paper we prove which determines up to a factor of order , and solves a conjecture of Erdős.
Paper Structure (13 sections, 10 theorems, 54 equations)

This paper contains 13 sections, 10 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Unitals and the O'Nan configuration
  4. The graph $H_q$
  5. Clique structure of $H_q$
  6. The random $K_4$-free graph $H_q^*$
  7. Pseudorandomness in $H_q^*$
  8. Proof of Theorem \ref{['thm:pseudo']}
  9. Randomly sampling from $G_q^*$
  10. Counting independent sets
  11. The proof of Theorem \ref{['thm:main']}
  12. Proof of Theorem \ref{['thm:main2']}
  13. Concluding remarks

Key Result

Theorem 1

As $t \rightarrow \infty$,

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Proposition 3
  • Lemma 3
  • Proposition 4
  • ...and 1 more