Prove that the Ramsey number $R(4,4;3) = 13$

Question

Prove that the Ramsey number $R(4,4;3) = 13$.
I don't know how to deal with the Ramsey number $R(p_1,p_2,...,p_k;r)$ where r is larger than 2.
Is there any useful inequality or construction of graph?

Answer 1

This appears to be a very hard problem, but I can give you a reference to the paper in which it was solved.

That result appears in Brendan D. McKay & Stanisław P. Radziszowski, The First Classical Ramsey Number for Hypergraphs Is Computed, which appeared in $1991$; a PDF can be freely downloaded from here. According to the paper’s introduction, Isbell had presented an example in $1969$ to show that $R(4,4;3)\ge 13$; the reference is given. McKay and Radziszowski used a variety of theoretical arguments to narrow the search space and then used a computer to show that $R(4,4;3)\le 13$, thereby establishing the result.

Answer 2

I noticed that I am probably in the same Graph Theory class with you. After some surveying, I found that this survey, Small Ramsey Number, which was published in 2014, stated that the computer-aided proof back in 1991 was the newest.

(Not enough reputation to comment, sorry for not much help provided)