Prime counts with maximal prime spacing. I think this question is related to Maier's theorem but I am unsure.
Notation: $\pi(x)$ is the prime-counting function up to $x$. $g_k := p_{k+1} - p_k$.
Define $M_n$ to be the $n$th maximal gap between primes. That is, $M_n = g_i$, where $p_i$ is the smallest prime such that $g_i > g_j$ for all $j < i$. Define $x_n := $smallest $x$ where $m_n := max(\pi(x+M_n)-\pi(x))$ for all $x$ between $p_i$ of $M_{n-1}$ to the $p_i$ of $M_n$ occurs.
In other words, using the spacing between maximal primes as a section, what is the maximum count of primes for each section? And, where is it? Does the count $m_n$ go to infinity? At what rate does it grow? Clearly, $\pi(x+M_n)-\pi(x)$ is greater than 1 up to the maximal prime where is it 1. What happens with $m_n/M_n$?
 A: Does the count $m_n$ go to infinity? At what rate does it grow?
Heuristically, yes, $m_n$ goes to infinity; assuming that the Cramer and Shanks conjectures are true for almost all maximal prime gaps, we can expect that $m_n$ goes to infinity at least as fast as $\log x_n$. That's because 
(i) we almost always have $M_n\sim\log^2 x_n$ by the Cramer and Shanks conjectures;
(ii) the average count of primes in the interval $[x,x+M_n]$ (for $x$ near $x_n$) is estimated by the ratio 
$${M_n\over\mbox{average gap near $x_n$}}\sim {\log^2 x\over\log x}
\sim\log x \sim\log x_n;
$$ 
(iii) it's intuitively clear that we will almost always see $m_n$ exceeding the average count of primes in $[x,x+M_n]$ (for $x$ near $x_n$). That is,
$$
m_n > \log x_n \qquad\mbox{ for almost all } n.
$$
What happens with $m_n/M_n$?
We can expect that the ratio $m_n/M_n$ is bounded. If the ratio decreases to zero, then the decrease is surely not as fast as $1/\log x_n$. (But I would be very surprised to see a rigorous proof of this.) 
