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Sharp collision criteria for frozen color histograms

Scott Schubr

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Abstract

A probability law selects fixed-size subsets of a finite ground set, whose elements are independently colored once. We compare the histogram law conditional on that frozen coloring with the prescribed multinomial law. Under upper negative-cylinder inequalities, the normalized marginal collision sum gives a sharp criterion for expected chi-square convergence and its leading coefficient, uniformly over positive varying color probabilities and color counts. For a fixed positive color vector, the same collision sum characterizes expected total-variation convergence; necessity holds for every fixed-size source law. A non-real-stable example and applications to spanning trees and matroid-base rounding illustrate the scope.

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Edition files

Available editions

  1. Edition 3 · 2026-11-02 · public
  2. Edition 2 · 2026-10-27 · public
  3. Edition 1 · 2026-10-26 · public

Research details

Subjects
Mathematics · Combinatorics and Graph Theory · Probability and Statistics
Keywords
multinomial approximation, negative dependence, frozen coloring, collision probability, total variation, matroid rounding
Manuscript date
2026-10-26
Reader reuse
No additional permission granted

Scope

Under upper negative-cylinder inequalities, (m−1)τ characterizes expected chi-square convergence uniformly over positive color probabilities; fixed positive colors give a τ criterion for expected total variation, with a necessary covariance bound for every fixed-size subset law.

Disclosures

Author-posted preprint. OpenAI ChatGPT and Codex assisted exploration, literature comparison, drafting, derivation, and internal proof and code checks, as disclosed in the manuscript. The author assumes responsibility for all content.

Reproduction

Nine-page analytical paper and eleven-member package with complete editable source, inline bibliography, portable Tectonic build, both exact standard-library checkers and actual deterministic outputs. Normal and optimized controls agree; universal and asymptotic results are proved analytically. Archive edition 2.

References and related resources

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