Sunflowers in Set Systems with Small VC-Dimension
Bibliographic record
Abstract
Abstract A family of r distinct sets $$\{A_1,\ldots , A_r\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is an r -sunflower if for all $$1 \leqslant i < j\leqslant r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>i</mml:mi> <mml:mo><</mml:mo> <mml:mi>j</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> and $$1 \leqslant i' < j'\leqslant r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⩽</mml:mo> <mml:msup> <mml:mi>i</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo><</mml:mo> <mml:msup> <mml:mi>j</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>⩽</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> , we have $$A_i\cap A_j = A_{i'}\cap A_{j'}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>i</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>j</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> </mml:mrow> </mml:math> . Erdős and Rado conjectured in 1960 that every family $$\mathcal {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> of $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -element sets of size at least $$K(r)^\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ℓ</mml:mi> </mml:msup> </mml:mrow> </mml:math> contains an r -sunflower, where K ( r ) is some function that depends only on r . We prove that if $$\mathcal {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> is a family of $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -element sets of VC-dimension at most d and $$|\mathcal H| > (C r(\log d+\log ^*\ell ))^\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>H</mml:mi> <mml:mo>|</mml:mo> <mml:mo>></mml:mo> </mml:mrow> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mo>log</mml:mo> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ℓ</mml:mi> </mml:msup> </mml:mrow> </mml:math> for some absolute constant $$C > 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".