I don't think asymptotic estimates of that form suffice to treat this problem. (Where else in combinatorics has an argument of this form succeeded? What intuitive reason is there to expect it to succeed here?)

Specifically I think section 4 is basically nonsense. (I see Sniffnoy has already pointed this out below.)

(Re: your comment, Theorem 7 is going to fail below the smallest counterexample, right? This is bad, imprecise writing - a red flag.)