Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$K_4^-$-free triple systems without large stars in the complement

Dhruv Mubayi, Nicholas Spanier

TL;DR

This paper resolves the hypergraph Ramsey problem for the smallest nontrivial 3-graph, showing $r(K_4^-,S_n)=\Theta\left(\frac{n^2}{\log n}\right)$. It achieves this by constructing a $K_4^-$-free 3-graph via a nibble process that preserves strong pseudorandomness and forbids large stars in the complement; the approach is guided by a differential-equation heuristic tracking $q_i$ and $\pi_i$, culminating in a final graph where every vertex’s link is triangle-free. The authors develop a suite of concentration tools (Chernoff, McDiarmid-type, Warnke, Krivelevich) to control complex dependencies arising in the semi-random edge-selection and removal steps, and they prove that the desired edge-distribution property holds whp for all disjoint $n$-sets $A,B$ and external vertex $x$. The result generalizes Kim’s graph bound and confirms a conjecture linking polynomial Ramsey growth to a restricted class of hypergraphs, highlighting a distinct hypergraph phenomenon where link structures enforce stronger lower bounds. Overall, the work advances the understanding of Ramsey phenomena in hypergraphs by combining a delicate nibble construction with robust probabilistic concentration in a pseudorandom regime.

Abstract

The $n$-star $S_n$ is the $n$-vertex triple system with ${n-1 \choose 2}$ edges all of which contain a fixed vertex, and $K_4^-$ is the unique triple system with four vertices and three edges. We prove that the Ramsey number $r(K_4^-, S_n)$ has order of magnitude $n^2 /\log n$. This confirms a conjecture of Conlon, Fox, He, Suk, Verstraëte and the first author. It also generalizes the well-known bound of Kim for the graph Ramsey number $r(3,n)$, as the link of any vertex in a $K_4^-$-free triple system is a triangle-free graph. Our method builds on the approach of Guo and Warnke who adapted Kim's lower bound for $r(3,n)$ to the pseudorandom setting.

$K_4^-$-free triple systems without large stars in the complement

TL;DR

This paper resolves the hypergraph Ramsey problem for the smallest nontrivial 3-graph, showing . It achieves this by constructing a -free 3-graph via a nibble process that preserves strong pseudorandomness and forbids large stars in the complement; the approach is guided by a differential-equation heuristic tracking and , culminating in a final graph where every vertex’s link is triangle-free. The authors develop a suite of concentration tools (Chernoff, McDiarmid-type, Warnke, Krivelevich) to control complex dependencies arising in the semi-random edge-selection and removal steps, and they prove that the desired edge-distribution property holds whp for all disjoint -sets and external vertex . The result generalizes Kim’s graph bound and confirms a conjecture linking polynomial Ramsey growth to a restricted class of hypergraphs, highlighting a distinct hypergraph phenomenon where link structures enforce stronger lower bounds. Overall, the work advances the understanding of Ramsey phenomena in hypergraphs by combining a delicate nibble construction with robust probabilistic concentration in a pseudorandom regime.

Abstract

The -star is the -vertex triple system with edges all of which contain a fixed vertex, and is the unique triple system with four vertices and three edges. We prove that the Ramsey number has order of magnitude . This confirms a conjecture of Conlon, Fox, He, Suk, Verstraëte and the first author. It also generalizes the well-known bound of Kim for the graph Ramsey number , as the link of any vertex in a -free triple system is a triangle-free graph. Our method builds on the approach of Guo and Warnke who adapted Kim's lower bound for to the pseudorandom setting.

Paper Structure

This paper contains 20 sections, 13 theorems, 198 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Nibble Algorithm
  3. Expected Trajectories via Differential Equations
  4. Technical Estimates
  5. Events
  6. Concentration bounds
  7. Proof of Lemma \ref{['secondary_lemma']}
  8. Bound on $\neg \mathcal{N}$
  9. Bound on $\neg \mathcal{P}$
  10. Bound on $\neg \mathcal{P}^+$
  11. Proof of $\mathcal{N}^+$
  12. Proof of $\mathcal{Q}^+$
  13. Proof of $\mathcal{Q}$
  14. Proof of Lemma \ref{['Q event lemma']}
  15. $X_1$
  16. ...and 5 more sections

Key Result

Theorem 1

There is an absolute constant $c$ such that for all sufficiently large $N$, there is a $K_4^-$-free 3-graph on $N$ vertices whose complement contains no copy of $S_n$, where $n = c\sqrt{N\log N}$. In other words, $r(K_4^-, S_n) = \Omega(n^2/\log n)$.

Figures (3)

  • Figure 1: $T_i(x,uv) \text{, } S_i(x,uv) \text{, and } R_i(x,uv)$
  • Figure 2: $U_i(x,u,v,w)$
  • Figure 3:

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2: Main result
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6: McDiarmid1998Warnke
  • Theorem 7: McDiarmid1989McDiarmid1998Warnke
  • Theorem 8: see Warnke2017
  • Theorem 9: Krivelevich
  • Lemma 10
  • ...and 6 more