Sharp logarithmic exponents for off-diagonal Ramsey numbers

1 · The essentials

Imagine a collection of points, with some pairs joined and others not. How many points force either a group of five in which every pair is joined, or a group of a chosen size in which no pair is joined? The smallest number that guarantees this is called $r (5, t)$ , where $t$ is the chosen size of the second group. The main manuscript determines how this number grows as $t$ becomes very large, including the precise power of a logarithm that appears in its description. A companion manuscript gives the corresponding result when five is replaced by any larger fixed whole number. Together, these results match a previously known upper bound at the level of that logarithmic power, rather than giving an exact formula for every group size. The main result has a Lean formalization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, think of a party where each pair of guests either knows one another or does not. The question asks how large the party must be to guarantee either five mutual acquaintances or a chosen number of mutual strangers, regardless of who knows whom.

Why it matters

The logarithmic factor refines the growth rate beyond the power of $t$ alone. The manuscripts determine its exponent for every fixed integer $s \geq 5$ , matching the classical upper bound at that scale. This identifies a precise feature of the threshold without claiming an exact finite-size formula.

2 · Going further

The off-diagonal Ramsey number $r (s, t)$ is the least number of vertices that forces either an $s$ -clique or an independent set of size $t$ . The problem here keeps $s$ fixed while $t$ tends to infinity. The repository summary identifies a classical upper bound and says that the reported results match it at the scale of the logarithmic exponent.

The main manuscript states $r (5, t) = t^{4} / (lo g t)^{3 + o (1)}$ as $t \to \infty$ . Here $o (1)$ denotes a quantity tending to zero, so the exponent in the denominator approaches $3$ . The result specifies that limiting exponent, not an exact expression with denominator precisely $(lo g t)^{3}$ .

The companion manuscript states that, for every fixed integer $s \geq 6$ , $r (s, t) = t^{s - 1} / (lo g t)^{s - 2 + o (1)}$ as $t \to \infty$ . Together with the main manuscript's case $s = 5$ , this gives the family described in the repository summary. The supplied abstracts state the conclusions but do not describe the argument or construction, so an account of the proof strategy cannot be extracted from them. The main result has a Lean formalization; the supplied material does not specify its detailed scope or establish formalization of every companion claim.

Useful notions

Graphs: vertices and edges
Cliques and independent sets
Logarithms and powers
Limits and asymptotic notation

What was known before / what is new

Before

The repository summary refers to a classical upper bound for these Ramsey numbers. It says the new formulas match that bound at the logarithmic-exponent scale. The supplied text gives no further historical context or description of earlier lower bounds.

New

The main manuscript determines the logarithmic exponent $3$ for $r (5, t)$ . The companion manuscript reports the exponent $s - 2$ for every fixed integer $s \geq 6$ . Together, their statements cover every fixed integer $s \geq 5$ .

Limits and assumptions

The statements assume that $s$ is a fixed integer at least $5$ and that $t \to \infty$ ; they do not state a result for $s$ growing with $t$ . The $o (1)$ term allows a vanishing error in the logarithmic exponent, so the formulas do not supply exact finite values or a fixed multiplicative constant. The main result has a Lean formalization, but its detailed scope is not supplied, and formal verification of the companion claims is not specified. The abstracts provide no argument details and identify no further open problems.

Mini glossary

Clique: A set of vertices in which every pair is joined by an edge.
Independent set: A set of vertices in which no pair is joined by an edge.
Off-diagonal Ramsey number: $r (s, t)$ is the least number of vertices forcing an $s$ -clique or an independent set of size $t$ .
Logarithmic exponent: The power to which a logarithm is raised, here the exponent of $lo g t$ in the denominator.
$o (1)$: A quantity that tends to zero in the stated limit.

Keywords : Off-diagonal Ramsey numbers · Logarithmic exponents · Asymptotic growth

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

For every fixed integer s ≥ 5, proves $r (s, t) = t^{s - 1} / (lo g t)^{s - 2 + o (1)}$ as $t \to \infty$ , determining the logarithmic exponent and matching the classical upper bound at that scale. Here $r (s, t)$ is the least number of vertices forcing an s-clique or a t-vertex independent set.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/170.md

Manuscripts in this family

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The sharp logarithmic exponent of r(5,t)

24/09/2026

To be verified

We determine the sharp logarithmic exponent of the off-diagonal Ramsey number $r (5, t)$ : $r (5, t) = \frac{t ^{4}}{( lo g t ) ^{3 + o (1)}} (t ⟶ \infty) .$

PDF GitHub folder

Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers

24/09/2026

To be verified

For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number: $r (s, t) = \frac{t ^{s - 1}}{( lo g t ) ^{s - 2 + o (1)}} (t ⟶ \infty) .$

PDF GitHub folder