Proves r(s, k) ≥ Ω(k^{s-1} / (log k)^{2s-4}) for fixed s ≥ 3 and k → ∞, nearly matching the Erdős-Szekeres upper bound and improving the Spencer lower bound for s ≥ 5.
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13 Pith papers cite this work, alongside 2 external citations. Polarity classification is still indexing.
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Establishes the first exponential improvement since 1947 to the lower bound on off-diagonal Ramsey numbers r(ℓ, Cℓ) for constant C > 1.
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.
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.
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.
f^{(4)}_{5^{-},6}(N) equals (log log N) to the Theta(1) power, with improved lower bounds r_4(6,n) >= 2^{2^{c sqrt(n)}} and r_k(k+2,n).
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).
New iterative proof of Nikiforov's theorem on H-blowups that improves the constant c_H(γ).
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.
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.
Simplified proof of exponential Ramsey lower bound improvements via Gaussian random graphs, with better quantitative constants than prior work.
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.
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.
citing papers explorer
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Off-diagonal Ramsey numbers
Proves r(s, k) ≥ Ω(k^{s-1} / (log k)^{2s-4}) for fixed s ≥ 3 and k → ∞, nearly matching the Erdős-Szekeres upper bound and improving the Spencer lower bound for s ≥ 5.
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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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A double-exponential lower bound for $r_4(5,n)$
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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A Note on Generalized Erd\H{o}s-Rogers Problems
f^{(4)}_{5^{-},6}(N) equals (log log N) to the Theta(1) power, with improved lower bounds r_4(6,n) >= 2^{2^{c sqrt(n)}} and r_k(k+2,n).
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A linear upper bound for zero-sum Ramsey numbers of bounded degree graphs
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
Simplified proof of exponential Ramsey lower bound improvements via Gaussian random graphs, with better quantitative constants than prior work.
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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 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.