Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A step towards the Erdős-Rogers problem

Longma Du, Xinyu Hu, Ruilong Liu, Guanghui Wang

Abstract

For $2\le k\le t<s$, the Erdős-Rogers function $f^{(k)}_{t,s}(N)$ denotes the largest $m$ such that every $K^{(k)}_s$-free $k$-graph on $N$ vertices contains a $K^{(k)}_t$-free induced subgraph on $m$ vertices. Mubayi and Suk (J. London Math. Soc. 2018) conjectured that $f^{(k)}_{k+1,k+2}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for $k\ge 4$, where $\log_{(i)}$ denotes the $i$-fold iterated logarithm. This is equivalent to the statement that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for every $s\ge k+2$. In this paper, we introduce multi-color patterns into a random construction of a $2$-graph to build a $4$-graph, and for the first time, combine them with multi-layer extremum structures to prove that $f^{(4)}_{5,s}(N)=(\log \log N)^{Θ(1)}$ for every $s\ge 11$. More generally, using a variant of the Erdős-Hajnal stepping-up lemma, we also establish that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for every $s\ge k+7$.

A step towards the Erdős-Rogers problem

Abstract

For , the Erdős-Rogers function denotes the largest such that every -free -graph on vertices contains a -free induced subgraph on vertices. Mubayi and Suk (J. London Math. Soc. 2018) conjectured that for , where denotes the -fold iterated logarithm. This is equivalent to the statement that for every . In this paper, we introduce multi-color patterns into a random construction of a -graph to build a -graph, and for the first time, combine them with multi-layer extremum structures to prove that for every . More generally, using a variant of the Erdős-Hajnal stepping-up lemma, we also establish that for every .
Paper Structure (7 sections, 5 theorems, 22 equations, 2 figures)

This paper contains 7 sections, 5 theorems, 22 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Properties of the stepping-up technique
  3. The upper bound for $f^{(4)}_{5,s}(N)$ with $s\ge 11$
  4. $H$ is $K_s^{(4)}$-free for every $s\ge 11$
  5. $\alpha_5(H)<2^{11}n^{11}+1$
  6. Proof of Corollary \ref{['main-results-2']}
  7. Concluding Remarks

Key Result

Theorem 1.2

For every $s\ge 11$, we have $f^{(4)}_{5,s}(N)= (\log\log N)^{\Theta(1)}$.

Figures (2)

  • Figure 1: A diagram illustrating the proof process for Claims \ref{['blue']} and \ref{['green']}.
  • Figure 2: The process of finding $K_5^{(4)}$ based on Claim \ref{['blue']} and Claim \ref{['green']}.

Theorems & Definitions (12)

  • Conjecture 1.1: Rephrasing of a conjecture of Mubayi and Suk in M-S-5
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 3.1
  • Claim 3.2
  • Claim 3.3
  • Claim 3.4
  • Lemma 3.5
  • Claim 3.7
  • Claim 3.8
  • ...and 2 more