Some negative answers to Bergelson–Hindman’s question

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