On the Gap Between Hereditary Discrepancy and the Determinant Lower Bound
Bibliographic record
Abstract
.The determinant lower bound of Lovász, Spencer, and Vesztergombi [European J. Combin., 7 (1986), pp. 151–160] is a general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovász, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the determinant lower bound. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of \(m\) subsets of a universe of size \(n\) is on the order of \(\max \{\log n,\sqrt{\log m}\}\). On the other hand, building upon work of Matoušek [Proc. Amer. Math. Soc., 141 (2013), pp. 451–460], Jiang and Reis [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp. 308–313] showed that this gap is always bounded up to constants by \(\sqrt{\log (m)\log (n)}\). This is tight when \(m\) is polynomial in \(n\) but leaves open the case of large \(m\). We show that the bound of Jiang and Reis is tight for nearly the entire range of \(m\). Our proof amplifies the discrepancy lower bounds of a set system derived from the discrete Haar basis via Kronecker products.Keywordsdiscrepancy theorylinear algebraincidence matricesHaar basisdeterminantsMSC codes05B2011K3805D05
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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 | 0.006 | 0.048 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.003 | 0.007 |
| Scholarly communication | 0.004 | 0.013 |
| Open science | 0.003 | 0.008 |
| Research integrity | 0.002 | 0.008 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".