Some negative answers to Bergelson–Hindman’s question
Volume 271 / 2025
Fundamenta Mathematicae 271 (2025), 227-236
MSC: Primary 05D10
DOI: 10.4064/fm240922-24-8
Published online: 16 October 2025
Abstract
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 “$\varDelta ^*$”.
