I am a graduate student trying to get involved in Ramsey theory. My question comes from:
Erdős on graphs: his legacy of unsolved problems By Fan R. K. Chung, Paul Erdős, Ronald L. Graham
p.14 of this book is available as a google ebook.
They quote Erdos in a 1980/1981 paper
"Faudree, Shelp, Rousseau, and I needed recently a lemma stating:
(R(n+1,n)-R(n,n))/n --> infinity as n--> infinity. We could prove this without much difficulty."
My first question is how does one show this?? I have tried my hand at this using recursive formulas for bounds on ramsey numbers, but it seems the lower bounds of this type are particularly weak. Furthermore any proof of this is absent from the literature to my knowledge. Any insight will be very very much appreciated.
Also which is larger R(k,k-2)+1 or R(k-1,k-1)? Of course the second but can we prove it?
Also for m+n=r+s=v with m < r =< v/2 =< s < n can we prove R(r,s) >= R(m,n) ?
Bounty to the best response :)