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Later they moved to Australia. They passed away on the same day Aug. 28, 2008, within one hour of each other. 54 Proof. (second proof of E-S) We claim N = R(k, 5; 4) is big enough. , we colour a 4 point sets as Y if they are convex, B if they are not. 16). I don’t feel the least humble before the vastness of the heavens. The stars may be large, but they cannot think or love; and these are qualities which impress me far more than size does. – Frank P. Ramsey (1903 - 1930) 55 11 The basic probabilistic method The first problem in this section is a cute result from Erd¨os.

And r(y, b) ≤ r(y − 1, b) + r(y, b − 1). 7. r(y, b) ≤ y+b−2 y−1 . In particular, r(k, k) ≤ 2k−2 k−1 . The proof is left as an exercise. We will prove the more general Ramsey’s theorem. Clearly r(k, l) = r(l, k). S. r(2, k) = k. r(3, 3) = 6, r(3, 4) = 9 as we proved. r(3, 5) = 14, r(3, 6) = 18, r(3, 7) = 23, r(3, 8) = 28, r(3, 9) = 36. r(4, 4) = 18 as we proved. r(4, 5) = 25. These are all the Ramsey numbers we know. r(5, 5) is between 43 and 49, inclusive. 1. Find, or improve the bound for, any unknown Ramsey number.

N log 2 n log 2(n)r = < n log 2 p (⌊n/2⌋)r n−r+1 n/2 − r + 1/2 r < 2n2r (1 + r )r . n − 2r + 1 Note that (1 + 1/r)r → e, it is easy to see the last quantity is optimized when n is in the order of r 2 , and we have m(r) ∈ O(r 22r ). 12 (Erd¨os 1964). m(r) < (1 + o(1))e(log 2)r 22r−2 . For the specific values, we only know m(r) for r ≤ 3. m(4) is between 20 and 23. 1 (Erd¨os - Lov´asz). m(r) = Θ(r2r ). J. Beck improved m(r) to Ω(r 1/3 2r ) in 1978 using alterations, based on his proof, the lower bound was improved in 2000 with some more tricks.