  Following their resolution of the Erd\H{o}s $B+B+t$ problem, Kra Moreira,
Richter, and Robertson posed a number of questions and conjectures related to
infinite configurations in positive density subsets of the integers and other
amenable groups. We give a negative answer to several of these questions and
conjectures by producing families of counterexamples based on a construction of
Ernst Straus.
  Included among our counterexamples, we exhibit, for any $\varepsilon > 0$, a
set $A \subseteq \mathbb{N}$ with multiplicative upper Banach density at least
$1 - \varepsilon$ such that $A$ does not contain any dilated product set
$\{b_1b_2t : b_1, b_2 \in B, b_1 \ne b_2\}$ for an infinite set $B \subseteq
\mathbb{N}$ and $t \in \mathbb{Q}_{>0}$. We also prove the existence of a set
$A \subseteq \mathbb{N}$ with additive upper Banach density at least $1 -
\varepsilon$ such that $A$ does not contain any polynomial configuration
$\{b_1^2 + b_2 + t : b_1, b_2 \in B, b_1 < b_2\}$ for an infinite set $B
\subseteq \mathbb{N}$ and $t \in \mathbb{Z}$. Counterexamples to some closely
related problems are also discussed.
