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Math-2622 · mathematics · Combinatorics

Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed integer $s>=5$ and every $\epsilon >0$, is r(s,t) bounded below by $t^{s-1}/(\log $ $t)^{s-2+\epsilon }$ for all sufficiently large t and above by $C_s$ $t^{s-1}/(\log $ $t)^{s-2}$, for a constant $C_s$ depending only on s?

Short English statement

For every fixed integer $s>=5$ and every $\epsilon >0$, is r(s,t) bounded below by $t^{s-1}/(\log $ $t)^{s-2+\epsilon }$ for all sufficiently large t and above by $C_s$ $t^{s-1}/(\log $ $t)^{s-2}$, for a constant $C_s$ depending only on s?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2622-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2622

The two selected manuscripts together cover the full fixed-size range $s\ge5$: one proves the displayed bounds for $s=5$, and the other for each $s\ge6$. The upper constant depends only on $s$, while the sufficiently-large-$t$ threshold may depend on $s$ and the positive lower-bound loss $\epsilon$.

Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers

OpenAI's claim

Ramsey logarithms for clique size at least six: For every fixed $s\ge6$, the manuscript claims $r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}$; the separate parameter $s=5$ is treated by another manuscript.

The logarithmic exponent of r$5,t$: The companion manuscript claims $r(5,t)=t^4/(\log t)^{3+o(1)}$; the lower estimate allows any arbitrarily small extra logarithmic power, while the matching upper estimate has power three.

Mathematical statement of the claimed result

Mathematical claim 1.
For every fixed integer $s\ge6$, the claimed bounds on the Ramsey number $r(s,t)$ are \[\frac{t^{s-1}}{(\log t)^{s-2+\epsilon}}\le r(s,t)\le C_s\frac{t^{s-1}}{(\log t)^{s-2}}\quad(\epsilon>0,\ t\ge t_0(s,\epsilon)).\] Here $r(s,t)$ is the least order forcing a clique of size $s$ or an independent set of size $t$. In particular $((s-1)\log t-\log r(s,t))/\log\log t\to s-2$. This paper covers $s\ge6$.
Mathematical claim 2.
The complementary fixed-clique-size result claims an absolute $C$ with \[\frac{t^4}{(\log t)^{3+\epsilon}}\le r(5,t)\le C\frac{t^4}{(\log t)^3}\quad(\epsilon>0,\ t\ge t_0(\epsilon)).\] Therefore $(4\log t-\log r(5,t))/\log\log t\to3$. Together with the separately cited $s\ge6$ manuscript, this covers every fixed $s\ge5$.

Outline of the argument

Ramsey logarithms for clique size at least six: Random incident point--hyperplane flags define an ordered graph in projective dimension $s-1$. Linear independence forbids the target clique, while independent sequences satisfy an asymmetric consistency rule. Sparse-incidence descriptions, validation samples, and dimensional projection give short encodings of such sequences, contradicting the entropy needed for a long independent set.

The logarithmic exponent of r$5,t$: Random flags in projective four-space produce a graph without five-cliques. Two marking scans and a small set of representative flags bound the information in a purported long consistent sequence. Geometric incidence analysis controls exceptional subspace rectangles, causing its upper entropy bound to conflict with the entropy forced by occurrence in the original random stream.

Formal verification

Family 170 describes Lean coverage for the Ramsey logarithmic exponents at clique size five and at every fixed clique size at least six.

Ramsey logarithms for clique size at least six: The manuscript links RamseyFive.lean, SharpLogRamsey.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

The logarithmic exponent of r$5,t$: The manuscript links RamseyFive.lean, SharpLogRamsey.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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