Threshold Constants for Asymmetric Random Ramsey Properties

Definitions and mathematical statement

Let $F_1,F_2$ be fixed finite simple graphs with \[ 1<m_2(F_1)\le m_2(F_2). \] For the standard two-density convention in this range, \[ m_2(F)=\max_{\substack{H\subseteq F\\v(H)\ge3}} \frac{e(H)-1}{v(H)-2}, \] where $v(H)=|V(H)|$ and $e(H)=|E(H)|$; smaller exceptional graphs do not determine the maximum under the displayed hypothesis. Retain the source's asymmetric density in its given order: \[ m_2(F_1,F_2)= \max_{\substack{H\subseteq F_2\\v(H)\ge2}} \frac{e(H)}{v(H)-2+1/m_2(F_1)}. \] Subgraphs are not required to be induced. The denominator is positive for every term under the hypothesis.

Let $G_{n,p}$ have vertex set $\{1,\ldots,n\}$ and independent edge probability $p$. Write $G\to(F_1,F_2)$ when every red/blue edge coloring of $G$ contains a red copy of $F_1$ or a blue copy of $F_2$. The source asks whether there is a constant $b>0$, depending on the fixed pair but not $n$, such that this Ramsey property has threshold \[ p=b\,n^{-1/m_2(F_1,F_2)}. \]

The word threshold is not specified precisely. In the usual threshold-function sense, for $q_n=n^{-1/m_2(F_1,F_2)}$ this would mean probability tending to zero when $p/q_n\to0$ and to one when $p/q_n\to\infty$; multiplying a threshold function by a fixed positive $b$ does not distinguish a constant in this sense. A sharper constant assertion would instead require, for each $\varepsilon\in(0,1)$, the probabilities at $(1-\varepsilon)bq_n$ and $(1+\varepsilon)bq_n$ to tend respectively to zero and one. The source does not select between these, a constant-factor zero/one statement with distinct constants, or a one-sided bound. No limiting probability at exactly $p=bq_n$ is supplied.

The exact density formula, ordering, and strict lower bound on $m_2(F_1)$ are preserved. No balancedness condition, graph-class restriction, or theorem about current random Ramsey thresholds is added.

Short English statement

The source proposes the displayed asymmetric density exponent for the random Ramsey threshold and asks for a positive threshold constant. Its exact density formula is explicit, but coarse threshold order and a sharp limiting constant are different meanings that the record does not distinguish.

Sources

• Local UnsolvedMath registry, permanent ID 30000609, code catalogue entry, "Threshold Constants for Asymmetric Random Ramsey Properties." Proposed year 2006; proposer unspecified.
• Combinatorics, Probability and Computing (2007), Oberwolfach Reports, pp. 2875--2952, DOI 10.4171/owr/2006/48; https://doi.org/10.4171/owr/2006/48. The displayed report year 2007 differs from the DOI and proposed year 2006. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied verification says official-report content was checked and notation, context, and question form normalized without changing the source claim.
• Literature triage checked 2026-08-21: open. It generically calls the extracted claim an unresolved problem/conjecture/direction and says the supplied check found no authoritative complete resolution. This does not identify which threshold notion was assessed. Created 2026-08-25; updated 2026-09-16. Source scope and current status require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.