Some negative answers to the Bergelson-Hindman's question

TL;DR

This paper provides counterexamples showing that certain polynomial-based combinatorial configurations do not necessarily exhibit largeness in various strong combinatorial senses, answering a question posed by Bergelson and Hindman negatively.

Contribution

It demonstrates that for integral polynomials vanishing at zero, the expected largeness properties in combinatorial sets do not always hold, countering previous conjectures.

Findings

Counterexamples for central* sets

Counterexamples for IP* sets

Counterexamples for Δ* sets

Abstract

Let $p_1,\dots,p_d$ be integral polynomials vanishing at $0$. It was asked by Bergelson and Hindman whenever $A$ is large, whether the set $\{(m,n)\in \mathbb{N}^2:m+p_1(n),m+p_2(n),\dots,m+p_d(n)\in A\}$ be large in the same sense. In this paper, we give negative answers to this question when ``large'' being the notions of ``central*'', ``IP*'', ``IP$_n$*'', ``IP$_{<\omega}$*'' and ``$\Delta$*''.