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Optimizing the CGMS upper bound on Ramsey numbers
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Optimizing the CGMS upper bound on Ramsey numbers
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In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the diagonal Ramsey numbers since 1935. We shorten their proof, replacing the underlying book algorithm with a simple inductive statement. This modification allows us - to give a very short proof of an improved upper bound on the off-diagonal Ramsey numbers, which extends to the multicolor setting, and - to clarify the dependence of the bounds on underlying parameters and optimize these parameters, obtaining, in particular, an upper bound $$R(k,k) \leq (3.8)^{k+o(k)}$$ on the diagonal Ramsey numbers.
Forward citations
Cited by 17 Pith papers
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Off-diagonal Ramsey numbers
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An exponential improvement for Ramsey lower bounds
Establishes the first exponential improvement since 1947 to the lower bound on off-diagonal Ramsey numbers r(ℓ, Cℓ) for constant C > 1.
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r_4(5,n) is at least 2^{2^{c n^{1/7}}}, determining the tower growth rate of r_k(k+1,n) for hypergraph Ramsey numbers.
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New Tower-Type Lower Bounds for Hypergraph Ramsey Numbers
Improves r_k(k+1,k+1) > s_3(⌊k/2⌋-2) for k≥6 and proves s_3(k) ≥ (twr_{k-2}(2))^2 for k≥5, yielding r_k(k+1,k+1) > (twr_{⌊k/2⌋-4}(2))^2 for k≥14.
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A degree version of the Burr-Erd\H{o}s conjecture on trees
Proves that graphs on N ≥ 2n vertices with δ(G) ≥ ⌊3N/4⌋ have every 2-edge-coloring containing a monochromatic copy of every n-vertex tree with max degree ≤ Δ.
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Ramsey numbers of multiple copies of a graph and the random Ramsey theorem
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A Note on Generalized Erd\H{o}s-Rogers Problems
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Zero-sum Ramsey numbers R(G, Γ) satisfy R(G, Γ) ≤ C n for bounded-degree n-vertex graphs G whenever |Γ| divides e(G).
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Finding blowups one vertex at a time
New iterative proof of Nikiforov's theorem on H-blowups that improves the constant c_H(γ).
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Ramsey numbers of multiple copies of a graph and the random Ramsey theorem
Multicolour Ramsey numbers for many copies of H are determined up to additive (or linear) error via (H,r)-gadgets, with random analogues generalising Rödl–Ruciński.
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New results on the odd- and unique-Ramsey numbers
New lower bounds r_odd(n, K_{s,t}) > n^{1/(s/2 + 1/(2 floor(t/8)))} for odd s even t, r_u(n, C_n) > n/4 creating a polynomial gap, and odd-Ramsey number of Hamilton cycles >1 in super-Dirac graphs.
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Gaussian random graphs and Ramsey numbers
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Sharper Ramsey lower bounds from refined Gaussian estimates
The exponent in the probabilistic lower bound for R(ℓ, Cℓ) is increased by a positive amount (asymptotically Θ(p_C^{-1/2}/log C) as C→∞) via a refined Gaussian estimate.
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Sharper Ramsey lower bounds from refined Gaussian estimates
The exponent in the lower bound for R(ℓ, Cℓ) increases by a positive amount for every fixed C>1, with asymptotic gain Θ(p_C^{-1/2}/log C) as C grows.
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An improved double-exponential lower bound for $r_4(5,n)$
The Ramsey number r_4(5,n) is at least 2^{2^{Omega(n^{1/5})}}, an improvement over the prior 2^{2^{Omega(n^{1/7})}} achieved by reducing greedy layers in the construction from seven to five.
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An improved double-exponential lower bound for $r_4(5,n)$
The paper establishes the improved lower bound r_4(5,n) >= 2^{2^{Omega(n^{1/5})}} for the 4-uniform 5-clique Ramsey number by reducing greedy local-maxima selection from seven layers to five in a modified construction.
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