The Erdős-Ko-Rado Theorem for 2-Pointwise and 2-Setwise Intersecting Permutations
Bibliographic record
Abstract
In this paper we consider the conjectured Erdős-Ko-Rado property for $2$-pointwise and $2$-setwise intersecting permutations. Two permutations $\sigma,\tau \in \operatorname{Sym}(n)$ are $t$-setwise intersecting if there exists a $t$-subset $S$ of $\{1,2,\dots,n\}$ such that $S^\sigma = S^\tau$. Further, two permutations $\sigma,\tau \in \operatorname{Sym}(n)$ are $t$-pointwise intersecting if there exists a $t$-subset $S$ of $\{1,2,\dots,n\}$ such that $s^\sigma = s^\tau$ for each $s \in S$. A family of permutations $\mathcal{F} \subset \operatorname{Sym}(n)$ is called $t$-setwise (resp. $t$-pointwise) intersecting, if any two permutations in $\mathcal{F}$ are $t$-setwise (resp. $t$-pointwise) intersecting. We say that $\operatorname{Sym}(n)$ has the $t$-setwise intersecting property if for any family $\mathcal{F}$ of $t$-setwise intersecting permutations, $|\mathcal{F}| \leqslant t!(n-t)!$. Similarly, $\operatorname{Sym}(n)$ has the $t$-pointwise intersecting property if for any family $\mathcal{F}$ of $t$-pointwise intersecting permutations, $|\mathcal{F}| \leqslant (n-t)!$.Ellis ([``"Setwise intersecting families of permutations". J. Combin. Theory Ser. A, 119(4):825-849, 2012]), proved that if $n$ is sufficiently large relative to $t$, then $\operatorname{Sym}(n)$ has the $t$-setwise intersecting property. Ellis also conjectured that this result holds for all $n \geqslant t$. Ellis, Friedgut and Pilpel ["``Intersecting families of permutations." J. Amer. Math. Soc. 24(3):649-682, 2011] also proved that for $n$ sufficiently large relative to $t$, $\operatorname{Sym}(n)$ has the $t$-pointwise intersecting property. It is also conjectured that $\operatorname{Sym}(n)$ has the $t$-pointwise intersecting property for $n\geqslant 2t+1$. In this work, we prove these two conjectures for $\operatorname{Sym}(n)$ when $t=2$.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 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".