Some negative answers to the Bergelson-Hindman's question

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 ``$\Delta$*''.