Saturation of \(K_{4}\) Subdivisions in Multidimensional Grids
Bibliographic record
Abstract
Let \(\mathcal{F}\) be a family of graphs, and \(H\) a ``host'' graph. A spanning subgraph \(G\) of \(H\) is called \(\mathcal{F}\)- saturated in \(H\) if \(G\) contains no member of \(\mathcal{F}\) as a subgraph, but \(G+e\) contains a member of \(\mathcal{F}\) for any edge \(e\in E(H) - E(G)\). We let \(Sat(H,\mathcal{F})\) be the minimum number of edges in any graph \(G\) which is \(\mathcal{F}\)-saturated in \(H\), where \(Sat(H,\mathcal{F}) = |E(H)|\) if \(H\) contains no member of \(\mathcal{F}\) as a subgraph. Let \(P_{m}^{r}\) be the \(r\)-dimensional grid, with entries in each coordinate taken from \(\{1,2,\cdots , m\}\), and \(K_{t}\) the complete graph on \(t\) vertices. Also let \(S(F)\) be the family of all subdivisions of a graph \(F\). There has been substantial previous work on extremal questions involving subdivisions of graphs, involving both \(Sat(K_{n},S(F))\) and the Turan function \(ex(K_{n},S(F))\), for \(F = K_{t}\) or \(F\) a complete bipartite graph. In this paper we study \(Sat(H, S(F))\) for the host graph \(H = P_{m}^{r}\), and \(F = K_{4}\), motivated by previous work on \(Sat(K_{n}, S(K_{t}))\). Our main results are the following; 1) If at least one of \(m\) or \(n\) is odd with \(m\geq 5\) and \(n\geq 5\), then \(Sat(P_{m}\times P_{n}, S(K_{4})) = mn + 1.\) 2) For \(m\) even and \(m\geq 4\), we have \(m^{3} + 1 \le Sat(P_{m}^{3}, S(K_{4}))\le m^{3} + 2.\) 3) For \( r\geq 3\) with \(m\) even and \(m\geq 4\), we have \(Sat(P_{m}^{r}, S(K_{4})) \le m^{r} + 2^{r-1} - 2\).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".