11.3: Estimating Ramsey Numbers
We will find it convenient to utilize the following approximation due to Stirling. You can find a proof in almost any advanced calculus book.
\(n! \approx \sqrt{2 \pi n} (\dfrac{n}{e})^n (1 + \dfrac{1}{12n} + \dfrac{1}{288n^2} - \dfrac{139}{51840n^3} + O(\dfrac{1}{n^4}))\)
Of course, we will normally be satisfied with the first term:
\(n! \approx \sqrt{2 \pi n} (\dfrac{n}{e})^n\)
Using Stirling's approximation and the binomial coefficients from the proof of Ramsey's Theorem for Graphs , we have the following upper bound:
\(R(n,n) \leq \dbinom{2n-2}{n-1} \approx \dfrac{2^{2n}}{4 \sqrt{\pi n}}\)