Erdős Magic
The twentieth century saw the elevation of Discrete Mathematics from ”the slums of topology” (one of the more polite expressions!) to its current highly regarded position in the mathematical pantheon. Paul Erd˝os played a key role in this transformation. We call discuss some key results, possibly including: i) Ramsey Theory. In 1946 Erdős showed that you could two-color the complete graph on n vertices so as to avoid a monochromatic clique of size k, where n was exponential in k. To do it, he introduced The Probabilistic Method. ii) Two-Coloring. In 1963 Erdős showed that given any m = 2 n − 1 sets, each of size n, one could two-color the underlying points so that none of the sets were monochromatic. His proof in two words: Color Randomly! There has been much work on larger m. We give a simple algorithm (together with a subtle analysis) of Cherkashin and Kozik that finds a coloring for the best known (so far!) m. iii) Number Theory. In 1940 Erdős , with Marc Kac, showed that the number of prime factors of n satisfies (when appropriately defined) a Gaussian distribution. Amazing! iv) Liar Games. Paul tries to find an integer from 1 to 100 by asking ten Yes/No questions from Carole. BUT, Carole can lie – though at most one time. Can Paul find the number? Or can Carole stop Paul with an Adversary Strategy? Anecdotes and personal recollections of Paul Erdős will be sprinkled liberally thoughout the presentation.
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Erdős Magic
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- 3. ¤ ¸ ½ n k 21−(k 2) 1¸ ¹ ¤ Kn Kkº ¿
- 4. ¸ ý ¹ ¸ ¸ ý ¸ ÎÁ ÎÁÁº ¸ ¸ ¹ ¸ ¸ ¹ ¸ ¹ º
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- 8. ¤ |Ai| = n¸ 1 ≤ i ≤ m = 2n−1k χ Ai ¤ ½ ¿℄ k 1 ⇒ ∃χ þ ¤ k ¤ ½ ℄ k = cn2¸ χ
- 9. m(n) = 2n−1k n¹ ý ½ ℄ k cn1/3 ⇒ ∃χ ¹ ¾¼¼¼℄ k c[n/ ln n]1/2 ⇒ ∃χ ¾¼½ ℄ ¹
- 10. þ v t(v) ∈ [0, 1]º v Á v ü ü ¹ e ÌÀ Æ ü v ü ÄË ü v ü f ü f v ¸ f t(v) = 1 2(1 + x)º e¸ v ¸ v ü ½¼
- 11. Á |x| pº x −pº þ w ∈ e ≤ 2−n(1 − p)n 2n−1k2−n(1 − p)n ü x +pº k(1 − p)n ÁÁ |x| ≤ pº (1 − x)n−1 w ∈ e (1 + x)n−1 z ∈ f [(1 − x)(1 + x)]n−1 ≤ 2−2(n−1) Pr[1 2(1 − p) ≤ t(v) ≤ 1 2(1 + p)] = p (2n−1k)2p2−2(n−1) = k2p k(1 − p)n + k2p ½½
- 12. k = k(n) ¸ p¸ 0 ≤ p ≤ 1¸ k(1 − p)n + k2 p ≤ 1 k(n) n → ∞º k ∼ c[n/ ln n]1/2 ¹ þ k, pº ü üý ü ¸ ¹ f þ ü þ ü ¤ ¹ ½¾
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- 14. ¤ ν(x) = ¹ x ¹ ½ ¾¼℄¸ ½ ¿ ℄ ý x ν(x) ∼ ln ln x x ∈ {1, . . . , n} º Xp p|x X = p≤n1/10 Xp ν(x) − 10 ≤ X(x) ≤ ν(x) E[Xp] = ⌊n p⌋/n = p−1 + O(n−1) E[X] = p−1 + o(1) = ln ln n + O(1) p, q Cov[Xp, Xq] = E[XpXq]−E[Xp]E[Xq] = O(n−1) ≤ n1/5 p = q¸ Cov[Xp, Xq] = o(1) ½
- 15. V ar[X] ≤ E[X] + Cov[Xp, Xq] ∼ ln ln n ´ ¹ ¤ µ Pr[|X − E[X]| K √ ln ln n] K−2 + o(1) ¤ ¹ Pr[|X−E[X]| K √ ln ln n] → K −∞ (2π)−1/2e−t2/2dt ½
- 16. ¹ ¸ ¹ º þ ¸ ¹ º º º ¤ ¹ ½½ ¸ ½ ¿ ½
- 17. ¹ º ¹ ¸ ¸ ¹ º ¹ ¸ ¹ ¹ º ¸ ¹ ¸ º ¤ ¸ ½
- 18. x ∈ {1, . . . , 100}º º º þ Pr[Ü ] = 11 1024 100 11 1024 1 1 ¸ ¤ ½
- 19. ¸ ¹ ¹ ¹ ¸ ¸ º ¹ ¸ ¸ ¸ º ¸ ¹ º ¹ ¸ ¸ ý ¹ º ý ¸ þ º ½
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