Enumeration and asymptotic analysis of edge-disjoint triangle packings in complete graphs
Bibliographic record
Abstract
<p>We investigate the combinatorial structure of edge-disjoint triangle packings in the complete graph <span class="math inline">\(K_n\)</span>. Two triangles are said to be edge-disjoint if they share no common edges, though they may share at most one vertex. For a given <span class="math inline">\(n\)</span>, let <span class="math inline">\(T_n\)</span> denote the total number of subsets of triangles in <span class="math inline">\(K_n\)</span> that are pairwise edge-disjoint, including the empty set, and let <span class="math inline">\(T_n^k\)</span> denote the number of <span class="math inline">\(k\)</span>-element sets of such triangles. In this article, we establish: (i) a general recurrence relation for <span class="math inline">\(T_n\)</span> that enables asymptotic analysis, yielding the growth <span class="math inline">\(\log T_n = \Theta(n^2 \log n)\)</span> for large <span class="math inline">\(n\)</span>; (ii) exact closed-form formulas for the number of edge-disjoint pairs (<span class="math inline">\(T_n^2\)</span>), triples (<span class="math inline">\(T_n^3\)</span>), and quadruples (<span class="math inline">\(T_n^4\)</span>) of triangles in <span class="math inline">\(K_n\)</span> for <span class="math inline">\(n \geq 6\)</span>. These results extend classical work on Steiner Triple Systems and provide new tools for analyzing triangle packings in complete graphs.</p>
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".