Reducing linear Hadwiger’s conjecture to coloring small graphs
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
In 1943, Hadwiger conjectured that every graph with no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript t"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">K_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> minor is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis t minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(t-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -colorable for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t\ge 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In the 1980s, Kostochka and Thomason independently proved that every graph with no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript t"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">K_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> minor has average degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis t StartRoot log t EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:msqrt> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>t</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(t\sqrt {\log t})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and hence is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis t StartRoot log t EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:msqrt> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>t</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(t\sqrt {\log t})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -colorable. Recently, Norin, Song and the second author showed that every graph with no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript t"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">K_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> minor is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis t left-parenthesis log t right-parenthesis Superscript beta Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> β </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(t(\log t)^{\beta })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -colorable for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 1 slash 4"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta > 1/4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , making the first improvement on the order of magnitude of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis t StartRoot log t EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:msqrt> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>t</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(t\sqrt {\log t})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bound. The first main result of this paper is that every graph with no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| 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 it