Local stability method for hypergraph Turán problems
Bibliographic record
Abstract
One of the earliest results in Extremal Combinatorics is Mantel's theorem from 1907 which says that the largest triangle-free graph on a given number of vertices is the complete bipartite graph with sizes of partition classes as equal as possible. In 1961 Turan asked the analogous question for 3-uniform hypergraphs - what is the largest 3-uniform hypergraph on a given vertex set with no tetrahedron? To this date, this number is unknown even asymptotically. Since the original question by Turan a new branch in Combinatorics, called hypergraph Turan-type problems, emerged. A typical Turan-type problem for an r-uniform hypergraph F asks for the maximum number of edges in an r-uniform hypergraph on given number of vertices without a copy of F; this number is called the Turan number of F. The major part of this thesis is devoted to such problems. In particular, we generalize and extend the classical stability method; a method pioneered by Erdos and Simonovits that is ubiquitous in the study of Turan-type problems. The developed method, referred as local stability method, is generically applicable and is of independent interest. In particular, it allows us to find new Turan numbers of several families of hypergraphs. Furthermore, we solve a conjecture of Frankl and Furedi from 1980's by determining the Turan number of a hypergraph called generalized triangle, for uniformities five and six. In the final part of the thesis we make some progress on one of the old conjectures of Erdos which states that every triangle-free graph on n vertices contains a subset of n/2 vertices that spans at most n^2/50 edges. We prove the conjecture under several natural assumptions, improving and generalizing previous results of of Keevash, Krivelevich and Sudakov.
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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 | 0.003 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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; both teacher heads agree on what is shown here.
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".