Supplementary Materials for "From Cell-Specific Heuristics to Transferable Structural Search for Ramsey Graph Construction"

This document provides supplementary experimental data, robustness analyses, sensitivity tests, and reviewer-requested evaluations supporting the manuscript: "From Cell-Specific Heuristics to Transferable Structural Search for Ramsey Graph Construction" submitted to MDPI Mathematics.

Supplementary content includes:

• Extended Robustness Results: Detailed performance metrics for all transfer and search methods across multiple random seeds, validating that the balanced-transfer story is not a single-seed artifact.
• Sensitivity Analysis: Evaluation under varying search budgets, exact supervision parameters, and structural resolution settings (compact and high-resolution distillation variants).

New Transfer Methods - Reviewer Additions:

• Hierarchical Coarsening Transfer: Reconstructs candidates by scaling concrete shift values from the most coarse (smallest-order) teacher witness proportionally to the target graph order, then lifting the coarse structure into the target space.
• Flag Algebra Density Transfer: A heuristic instantiation of flag-algebra local density matching - selects shifts greedily to match both edge density and triangle-clustering density observed across teacher witnesses.
  
  

Cross-Graph-Family and Scaling Experiments:

• Strict cross-r transfer: teacher pools drawn from a different clique-parameter family (r=3 teachers for r=4 targets), testing structure transfer across qualitatively different graph families.
• Minimal teacher representation: experiments under the most restricted teacher-pool budget, testing robustness when only compact structural priors are available.

Supplementary Figures and Tables: Visualizations of seed robustness, budget sensitivity, transfer sensitivity across scenarios, and portfolio stability.

The results validate the stability and transferability of the proposed structural search motifs in Ramsey graph construction, and demonstrate that portfolio_transfer maintains its top balanced rank when the two new methods are included in the comparison.