We can call the $x$th Fibonacci number Fib($x$). What's the best asymptotic lower bounds on the amount of relatively prime Fibonacci numbers between Fib($n$) and Fib($n+m$)?
In other words, if we take the $m$ Fibonacci numbers that lie between Fib($n$) (inclusive) and Fib($n+m$), what is the maximally sized set of these numbers can be pairwise relatively prime to each other, in terms of $n$ and $m$? Of course, I'm looking for some sort of asymptotic bounds - more specifically, a big Omega bound, but we are allowed to pick from any of the $m$ Fibonacci numbers in the sequence, in order to make this bound larger. Note that I'm looking for the maximally sized set, but in the worst case.