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arxiv: 2606.28216 · v1 · pith:QG443XX4new · submitted 2026-06-26 · 🧮 math.CO · math.PR

Fano Geometry and Slow Coupon Collecting

Dina Barak-Pelleg , Daniel Berend This is my paper

Pith reviewed 2026-06-29 03:07 UTC · model grok-4.3

classification 🧮 math.CO math.PR
keywords coupon collector problemfair mechanismsFano planeprojective planesexpected coverage timeMarkov chainsfinite geometrybalanced incomplete block designs
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The pith

The line set of the Fano plane defines a fair coupon collector whose expected coverage time exceeds that of the fully random model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the coupon collector problem where each draw must return a fixed number of coupons under a fairness requirement that every coupon appears equally often across all admissible draws. It directly disproves the Grunbaum-Yaakobi conjecture that the fully random model maximizes expected coverage time among all such fair mechanisms. The disproof rests on an explicit construction: the seven lines of the Fano plane supply admissible triples on seven coupons that remain fair yet produce a strictly larger expected time than uniform random triples. The same geometric approach extends to higher-order projective planes, while a separate infinite family called the star mechanism yields a closed-form expectation that can be either larger or smaller than the random case depending on scaling.

Core claim

We disprove the Grunbaum-Yaakobi conjecture by exhibiting the line set of the Fano plane as a fair mechanism whose expected coverage time exceeds that of the full model. Further exact and computational results are obtained for projective planes of higher order. In addition, we analyze a simple infinite family of fair mechanisms, the star mechanism, for which the expected coverage time admits a closed form. Depending on the scaling regime, this mechanism can be asymptotically slower or faster than the full model, showing that no universal extremality principle holds for fair mechanisms without additional structural assumptions.

What carries the argument

The line set of the Fano plane, which supplies the admissible draws while preserving fairness through balanced point-line incidence.

If this is right

  • The Grunbaum-Yaakobi conjecture is false.
  • Projective planes of order q supply fair mechanisms on q^{2} + q + 1 coupons.
  • The star mechanism admits an exact closed-form expression for expected coverage time.
  • No universal extremality principle holds for fair mechanisms without further structural assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Balanced incomplete block designs other than projective planes may furnish additional counterexamples.
  • The Markov-chain method used here extends naturally to any regular uniform hypergraph that meets the frequency-balance condition.
  • Applications that enforce fairness, such as randomized sampling protocols, may need to account for geometric constructions that increase rather than decrease expected completion time.

Load-bearing premise

The lines of the Fano plane satisfy the fairness condition that every coupon appears with identical frequency among the admissible draws, and the Markov-chain calculation of expected coverage time requires no post-hoc adjustments.

What would settle it

Direct computation of the absorption time of the coverage Markov chain for the seven-coupon Fano mechanism, compared against the known value 7 H_7 for the fully random model; equality or reversal would falsify the claim.

read the original abstract

We study the coupon collector's problem in a generalized setting where each draw reveals a fixed number of coupons and the sampling mechanism is required to be \emph{fair}, meaning that every coupon appears with the same frequency among the admissible draws. Grunbaum and Yaakobi conjectured that, among all fair mechanisms with fixed parameters, the fully random model maximizes the expected time to complete coverage. We disprove this conjecture by exhibiting explicit counterexamples arising from finite geometry. In particular, we show that the line set of the Fano plane yields a fair mechanism whose expected coverage time exceeds that of the full model. Further exact and computational results are obtained for projective planes of higher order. In addition, we analyze a simple infinite family of fair mechanisms, the star mechanism, for which the expected coverage time admits a closed form. Depending on the scaling regime, this mechanism can be asymptotically slower or faster than the full model, showing that no universal extremality principle holds for fair mechanisms without additional structural assumptions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper generalizes the coupon collector's problem to fair mechanisms (each coupon appears with equal frequency among admissible draws) and disproves the Grunbaum-Yaakobi conjecture that the fully random model maximizes expected coverage time. It exhibits an explicit counterexample using the lines of the Fano plane (a BIBD), shows this fair mechanism has strictly larger expectation than the uniform-over-all-triples model via absorbing Markov chain, provides exact/computational results for higher projective planes, and derives a closed form for the star mechanism family whose asymptotic behavior can be slower or faster than the full model.

Significance. If the explicit construction and calculations hold, the result is significant for supplying a concrete geometric disproof of the conjecture and for showing that no universal extremality principle exists for fair mechanisms. Strengths include the parameter-free explicit counterexample from finite geometry, the closed-form derivation for the star family, and the use of standard BIBD incidence counts together with the usual linear system for coupon-collector expectations.

major comments (1)
  1. The central disproof rests on the Fano-plane lines forming a fair collection and the Markov-chain expectation strictly exceeding the full model. The manuscript must exhibit the admissible 7 lines, confirm the balanced incidence (each point in exactly three lines), and display the solved linear system for the 2^7 state expectations to allow direct verification that the inequality holds without post-hoc state exclusions.
minor comments (2)
  1. Clarify the precise parameters (n=7, k=3) and the exact definition of fairness at the first use in the introduction.
  2. For the higher-order planes, state explicitly which orders were computed, the size of the state space, and any verification method used for the numerical expectations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. The suggestion to enhance the explicit verifiability of the Fano-plane counterexample is well-taken, and we will incorporate the requested details.

read point-by-point responses

  1. Referee: The central disproof rests on the Fano-plane lines forming a fair collection and the Markov-chain expectation strictly exceeding the full model. The manuscript must exhibit the admissible 7 lines, confirm the balanced incidence (each point in exactly three lines), and display the solved linear system for the 2^7 state expectations to allow direct verification that the inequality holds without post-hoc state exclusions.

    Authors: We agree that greater explicitness will strengthen the presentation. In the revised version we will list the seven admissible lines of the Fano plane explicitly, restate the BIBD parameters (v=7, k=3, λ=1) to confirm that each point lies in exactly three lines, and add an appendix containing the solved expectations for all 128 states of the absorbing Markov chain. The linear system was solved in full without any post-hoc exclusion of states; the appendix will make this computation directly checkable and will display the strict inequality between the Fano mechanism and the uniform model. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper disproves the Grunbaum-Yaakobi conjecture by explicit finite constructions (Fano plane lines as a fair mechanism) and standard absorbing Markov-chain expectation calculations on the power set. Fairness follows directly from the balanced incidence properties of the BIBD (projective plane of order 2), and the coverage-time comparison is obtained from the usual linear system without fitted parameters, self-referential equations, or load-bearing self-citations. No step reduces a claimed result to an input by definition or renames a known pattern; the derivation chain is self-contained against external combinatorial definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

With only the abstract available, no free parameters, ad-hoc axioms, or invented entities are visible; the work relies on standard Markov-chain expectation formulas for coupon-collector variants and the combinatorial definition of projective planes.

axioms (1)
  • standard math Standard properties of finite projective planes (every pair of points in exactly one line, etc.)
    Invoked to guarantee that the line set forms a fair mechanism.

pith-pipeline@v0.9.1-grok · 5693 in / 1134 out tokens · 66825 ms · 2026-06-29T03:07:07.775998+00:00 · methodology

discussion (0)

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Reference graph

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