  The Union-Closed Sets Conjecture, often attributed to P\'eter Frankl in 1979,
remains an open problem in discrete mathematics. It posits that for any finite
family of sets $S\neq\{\emptyset\}$, if the union of any two sets in the family
is also in the family, then $\underline{\text{there must exist an element that
belongs to at least half of the member sets}}$. We will refer to the underlined
text as the abundance condition. In 2001, David Reimer proved that the average
set size of a union-closed family $S$ must be at least
$\frac{1}{2}\log_{2}|S|$. When proving this result, he showed that a family
being union-closed implies that the family satisfies certain conditions, which
we will refer to as the Reimer's conditions. Therefore, as seen in the context
of Tim Gowers' polymath project on the Union-Closed Sets Conjecture, it is
natural to ask if all families that satisfy Reimer's conditions meet the
abundance condition. A minimal counterexample to this question was offered by
Raz in 2017. In this paper, we will discuss a general method to construct
infinitely many such counterexamples with any fixed lower bound on the size of
the member sets. Furthermore, we will discuss some properties related to these
counterexamples, especially those focusing on how far these counterexamples are
from being union-closed.
