1935 asymptotic base at selected λ
computing
computed from the entropy exponent
Ramsey upper bounds, operated from the exponent
The Four That Finally Moved
For decades, better diagonal Ramsey bounds shaved factors from around an exponential base of 4. In 2023, a theorem moved the base itself. Put the ratio on the diagonal and watch the browser assemble that peer-reviewed landmark from its published epsilon, then the stronger value from a later preprint.
Start with λThis page keeps finite bounds, asymptotic theorems, and scale illustrations in separate lanes.
Layer oneMove the exponent
Choose a red clique size k and a ratio λ = l/k. The exact finite lane uses integer l. The curve uses the continuous ratio from the asymptotic theorem.
Drag λ to 1. The three diagonal bases are drawn on a logarithmic vertical scale, because the first movement is too small for an honest ordinary bar chart.
live browser arithmeticUsed for the exact finite binomial and the scale-only ratio.
The exact lane rounds λk to an integer l and reports the actual ratio.
2023 explicit diagonal base
computing
published input: ε = 2^-7
2024 preprint base at selected λ
computing
includes a cited o(k) term
Exact finite Erdős-Szekeres bound, indexing convention R(k,l) ≤ C(k+l-2,l-1)
computing integer parameters
computing
Computing a text description of the logarithmic curve.
1935computing
2023computing
2024computing
The 2023 and 2024 expressions are asymptotic. This instrument does not turn either one into a certified upper bound at the finite k selected above.
Layer twoOpen the proof's local engine
A tiny change in the diagonal base can look cosmetic. The further result is structural: edit a finite colouring and test the exact excess-edge identity used by the later paper's candidate argument. This identity is a local shadow of the method, not a new proof of its asymptotic theorem.
Each cell is one edge from X to Y. Click to change red to blue. The paper measures red edges above a chosen baseline density p.
all 16 edges editablep is an exact multiple of 1/20 in this instrument.
Edges between X rows and Y columns
e_R(X,Y)
f_p(X,Y) = e_R - p|X||Y|
Σ_v f_p(X,N_R(v)∩Y)
p|X|f_p(X,Y)
Σ_y (deg_R(y,X)-p|X|)^2
left - base = remainder
computing degrees
Computing the identity.
The check
Every green value below is recomputed by this page from the current controls. Published constants enter only where they are named as published inputs.
The 2023 arithmetic
computing
The 2024 diagonal
computing
The exact recurrence
computing
The finite identity
computing
Dependencies and free choices
- Indexing: this page writes R(k,l) for a red K_k or blue K_l, and uses the finite recurrence bound C(k+l-2,l-1). The papers sometimes shift indices or abbreviate R(k,k) as R(k).
- Normalisation: every displayed base is per k. The entropy exponent is (1+λ)log(1+λ)-λlog λ, with natural logarithms.
- Finite versus asymptotic: the exact BigInt lane is finite. The base curves suppress subexponential terms. The pure-base ratio is only a scale illustration.
- Numerics: the binomial is exact BigInt arithmetic. Exponentials, logarithms, and the plotted curve use browser double precision.
- Controls: λ is a reader choice. For the finite lane, l is the nearest integer to λk. The graph partition has four vertices in X and four in Y. Its colouring and p are reader choices.
Unknown or not supplied
- The peer-reviewed Campos, Griffiths, Morris and Sahasrabudhe theorem says sufficiently large k but supplies no usable numerical threshold here.
- Sah's peer-reviewed pre-2023 bound uses an absolute c greater than zero. This page does not assign it a numerical value.
- The Gupta, Ndiaye, Norin and Wei formula contains o(k). Its size is not supplied by the asymptotic statement.
- The 2024 optimisation remains a preprint. As checked on 29 July 2026, arXiv lists only version 1 from 26 July 2024, and an author publication list gives no journal venue.
Still open
The exponential growth of diagonal Ramsey numbers is not determined. The cited lower and upper bounds leave an exponential gap. Neither the Annals theorem nor the later preprint determines exact large Ramsey numbers.
Offline differential check
The dependency-free verifier extracts these shipped functions, sweeps every reachable k and l, every λ step, and every one of the 65,536 matrix colourings at every p step against separately written references. It also mutates the extracted ε calculation in memory and requires that the tests catch the fault.
Primary sources
Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe, An exponential improvement for diagonal Ramsey, Annals of Mathematics 203(3), 869-932, May 2026. DOI 10.4007/annals.2026.203.3.4. The journal records acceptance on 30 June 2025 and online publication on 1 December 2025.
The same result at arXiv:2303.09521v2, first submitted 16 March 2023 and revised 4 August 2025. The manuscript supplies the explicit unoptimised choices 2^-10 and 2^-7. The second is used here.
Paul Erdős and George Szekeres, A combinatorial problem in geometry, Compositio Mathematica 2, 463-470, 1935. NUMDAM identifier CM_1935__2__463_0. Source of the classical finite recurrence bound.
Ashwin Sah, Diagonal Ramsey via effective quasirandomness, Duke Mathematical Journal 172(3), 545-567, 2023. DOI 10.1215/00127094-2022-0048. The theorem gives a subexponential improvement while retaining limiting base 4.
Parth Gupta, Ndiamé Ndiaye, Sergey Norin, and Louis Wei, Optimizing the CGMS upper bound on Ramsey numbers, arXiv:2407.19026v1, 26 July 2024. Source of G(λ), the 3.7992 diagonal value, and the excess identity used in layer two.
Lawrence C. Paulson, Formalising New Mathematics in Isabelle: Diagonal Ramsey, 16th International Conference on Interactive Theorem Proving, LIPIcs 352, 18:1-18:18, 2025. DOI 10.4230/LIPIcs.ITP.2025.18. This independently documents a formalisation of the 2023 result.