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.
Gaussian random graphs and Ramsey numbers
7 Pith papers cite this work. Polarity classification is still indexing.
7
Pith papers citing it
abstract
We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify their analysis significantly. As a consequence of this simpler analysis, we also obtain better quantitative bounds.
citation-role summary
background 2
citation-polarity summary
fields
math.CO 7years
2026 7verdicts
UNVERDICTED 7roles
background 2polarities
background 2representative citing papers
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.
The vertex-coloring coprime Ramsey number R_cop(k1,...,kc) equals the prime p indexed by sum(ki-1).
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).
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
-
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.
-
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.
-
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.
-
Prime Certificates for Exact Vertex-Coprime Ramsey Numbers
The vertex-coloring coprime Ramsey number R_cop(k1,...,kc) equals the prime p indexed by sum(ki-1).
-
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).
-
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.
-
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.