MathOverflow

Is there any relation between the incidence matrix of a 3 uniform linear hypergraphs and its kernel with the cycle space, and cyclomatic number of the hypergraph?

I have been exploring a variation of Brocard's problem ($1 + n! = m^2$). I decided to replace the constant $1$ with a prime number $p$, leading to the following equation:
$$p + n! = m^2$$
where $p$ is a prime, and $n, m$ are positive integers.

Lee and Li conjectured the following.
$10/8$-conjecture. If $X$ is a closed, oriented, smooth $4$-manifold with evenintersection lattice $aU \oplus bE_8$, where $U$ is the hyperbolic lattice, then $a \geq |b|$.
This is equivalent to $b_2(X) \geq \frac{10}{8}|\sigma(X)|$, where $\sigma$ is the signature, and hence the name. (Note that if $X$ is spin, Matsumoto's $11/8$-conjecture proposes an improvement by replacing $10/8$ with $11/8$.)...

I came back recently to a example due to the late Peter Nyikos which is written in an unfinished draft of his, "Antidiamond and anti-PFA axioms and topological applications", which is still available on his webpage (here: https://people.math.sc.edu/nyikos...