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Combinatorial Techniques and Objects in Computer Science: Fault-tolerance and Other Interesting Applications

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combinatorial-techniques-and-objects-in-computer-science-fault-2Number Theorymath.COmath.NTposed by N. M. Singhi, Iyengar Shriniwas, T. A. Antonyrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

Let S be the smallest family of subsets of I such that each t-subset of I occurs in at most \lambda blocks.Then S contains all subsets of size \geqslant(n-r') , where r' is the largest integer satisfying (n−tn−t)+(n−tn−t−1)+...+(n−tn−t−r′)⩽λ.\left(\begin{array}{c}n-t\\n-t \end{array}\right)+\left(\begin{array}{c}n-t\\n-t-1 \end{array}\right)+...+\left(\begin{array}{c}n-t\\n-t-r^{\prime}\end{array}\right)\leqslant \lambda.

Context

Candidate 2 of the open problems stated in "Combinatorial Techniques and Objects in Computer Science: Fault-tolerance and Other Interesting Applications", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: For a finite set II, ∣I∣=n|I|=n, integers 1≤t≤n1\le t\le n, and λ≥0\lambda\ge0, reconstruct the literal conjecture as follows: S⊆2IS\subseteq 2^I is a minimum-size family such that every tt-subset of II is contained in at most λ\lambda members of SS. If r′r' is the largest integer satisfying

    ∑j=0r′(n−tn−t−j)≤λ,\sum_{j=0}^{r'} \binom{n-t}{n-t-j}\le \lambda,

    then SS contains every subset of II of size at least n−r′n-r'.

    Result: The literal statement is false.

    Take

    I={1,2,3,4},t=2,λ=2.I=\{1,2,3,4\},\qquad t=2,\qquad \lambda=2.

    The empty family S=∅S=\varnothing satisfies the stated condition, since every 22-subset of II occurs in 0≤20\le2 blocks. It is also the unique smallest such family.

    Here n−t=2n-t=2, and

    (22)=1≤2,(22)+(21)=3>2,\binom22=1\le2,\qquad \binom22+\binom21=3>2,

    so r′=0r'=0. The claimed conclusion says that SS contains every subset of size at least 44, in particular II itself. But I∉∅I\notin\varnothing. Contradiction.

    Thus the conjecture as stated is disproved. The likely wording defect is “at most” instead of “at least”, since “minimal tt-cover” normally means every tt-subset is covered at least λ\lambda times. Even that universal repaired version is false: with the same I,t,λI,t,\lambda, the family of all four triples

    S={123,124,134,234}S=\{123,124,134,234\}

    covers every pair exactly twice and has minimum possible size 44, but it does not contain II, while again r′=0r'=0 would require I∈SI\in S.

    Citation: No known citation; this is a direct finite counterexample.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The proposed disproof gives a valid counterexample to the conjecture as literally stated. Under the “at most λ\lambda” condition, the empty family satisfies the requirement and is indeed the smallest family. For n=4,t=2,λ=2n=4,t=2,\lambda=2, one gets r′=0r'=0, so the conjectured conclusion would require I∈SI\in S, but S=∅S=\varnothing. This directly contradicts the stated conclusion.

      Novelty assessment

      TYPE1

      Classification rationale: The accepted resolution is a valid but very small finite counterexample to the literal wording. In fact, under the “at most λ\lambda” condition the empty family is automatically minimum, so the conjecture is vacuous/defective rather than a substantive extremal-combinatorics problem. This is not publishable as a standalone result; at most it would support an erratum or short comment.

      Literature check: I found no evidence that this exact counterexample or correction is recorded in the literature. Searches for “Conjecture Va.4.2”, “minimal t-cover” with Singhi, and the original paper title did not reveal an erratum or later paper resolving this literal statement. OpenAlex lists only a few citations to the original paper, notably Cooper–Ellis–Kahng on asymmetric binary covering codes, but nothing addressing this particular minimal tt-cover wording. No MathOverflow/GitHub-style traces were found either.

      Citation: N. M. Singhi, Iyengar Shriniwas, and T. A. Antony, “Combinatorial Techniques and Objects in Computer Science: Fault-tolerance and Other Interesting Applications,” European Journal of Combinatorics 17 (1996), 97–111, DOI: 10.1006/eujc.1996.0009.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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