I've been reading this article regarding Hawkes processes https://arxiv.org/pdf/1507.02822.pdf, and I've noticed that the definition for explosion in the article in page 3, stating that a counting process explodes if we have for $N(t)$ counting process, $t,s\in\mathbb{R}$ that: $$ N(t)-N(s)=\infty$$ That is, the probability of having infinite amount of "arrivals" in a finite interval is higher than $0$.
In page 8, the author says that given the ratio $n$, the process "explodes almost surely" for $n>1$. But, what he then shows is we have a series of arrivals $Z_i$ (You may view $Z_1$ as the first person and $Z_2$ as his son, and $Z_i$ being the $i$th son). Calculating the expectation of the amount of people we get that:
$$ \mathbb{E}\Big[\sum_{i=1}^\infty Z_i\Big]=\sum_{i=1}^\infty \mathbb{E}[Z_i]=\sum_{i=1}^\infty n^i = \infty$$
But, the author hasn't proved that the counting process explodes, he just proved that the expected amount of descendents is unbounded, but it does not mean that the process explodes (one might imagine a steadily exponentially increasing population, there need not be a finite time interval where we have infinite births).
Does it follow from this point and I just don't see it, or it is an extension of the definition of explosion of a counting process?
Not sure if this counts as a full answer given that I'm agreeing with you. But if I'm understanding things correctly then your idea of an exponentially increasing population over time actually seems pretty likely. I looked at the Asmussen reference ([24] in the paper), and if $1 < n < \infty$, then there should exist a $-\infty < \beta< 0$ such that $\int_{t > 0} e^{\beta t}\mu(t)\,dt = 1$.
Then Proposition V.7.1. of Asmussen states that $\lim_{t\to\infty}e^{\beta t}E[\lambda^*(t)] < \infty$. If the conditional intensity of the Hawkes process only increases exponentially fast we shouldn't expect explosion.
The only thing I can think of is that the authors were considering stationary distributions when they made that remark even though they they seem to be discussing the transient case up until that point. It's certainly true that if $n > 1$, the Hawkes process will not have a non-explosive, stationary distribution (assuming the constant $\lambda > 0$).