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A 910-block explicit construction guaranteeing a triple intersection with every $6$-subset of $[60]$

Paulo Henrique Cunha Gomes

TL;DR

This work presents an explicit construction of a 910-block family of 6-subsets on $[60]$ with the triple-intersection property, i.e., every 6-subset $S$ intersects some block in at least three elements, equivalently a radius-$3$ covering of the Johnson graph $J(60,6)$. The construction uses a simple two-halves split, internal pairings, and unions of three pairs to form blocks, yielding a concrete upper bound of $910$ for the covering number. A crude counting argument shows a general lower bound of $97$ blocks, and the paper discusses a generalization to $[2m]$ with $m$ even, along with an exploration of potential partition schemes and their limitations. Together with an analysis of limitations of certain splits and an alternative $(5,5)$ partition approach, the results illuminate explicit strategies for covering designs in Johnson graphs and provide a transparent benchmark for related combinatorial constructions and coding-theoretic contexts.

Abstract

We present a simple explicit family $\mathcal{B}$ of $910$ $6$-subsets of $[60]=\{1,\dots,60\}$ such that every $6$-subset $S\subset[60]$ intersects at least one block $B\in\mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$. Equivalently, $\mathcal{B}$ is a covering (dominating set) of the Johnson graph $J(60,6)$ with covering radius $3$ in the Johnson metric. The construction is purely combinatorial, based on a fixed split of $[60]$ into two halves, a pairing of each half, and a pigeonhole argument. We also record a crude counting lower bound and a straightforward generalization to $[2m]$ (with $m$ even).

A 910-block explicit construction guaranteeing a triple intersection with every $6$-subset of $[60]$

TL;DR

This work presents an explicit construction of a 910-block family of 6-subsets on with the triple-intersection property, i.e., every 6-subset intersects some block in at least three elements, equivalently a radius- covering of the Johnson graph . The construction uses a simple two-halves split, internal pairings, and unions of three pairs to form blocks, yielding a concrete upper bound of for the covering number. A crude counting argument shows a general lower bound of blocks, and the paper discusses a generalization to with even, along with an exploration of potential partition schemes and their limitations. Together with an analysis of limitations of certain splits and an alternative partition approach, the results illuminate explicit strategies for covering designs in Johnson graphs and provide a transparent benchmark for related combinatorial constructions and coding-theoretic contexts.

Abstract

We present a simple explicit family of -subsets of such that every -subset intersects at least one block in at least three elements, i.e.\ . Equivalently, is a covering (dominating set) of the Johnson graph with covering radius in the Johnson metric. The construction is purely combinatorial, based on a fixed split of into two halves, a pairing of each half, and a pigeonhole argument. We also record a crude counting lower bound and a straightforward generalization to (with even).
Paper Structure (11 sections, 4 theorems, 32 equations)

This paper contains 11 sections, 4 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Construction (910 blocks)
  3. Main result
  4. Lower bounds on the number of blocks
  5. Remarks and a generalization
  6. Integer solutions of $x_1+x_2+x_3=6$
  7. Why a $(3-blocks,3-blocks,4-blocks)$ split of the ten base blocks is not sufficient.
  8. A $(5-blocks,5-blocks)$ partition guarantees a triple after within-group recombination
  9. Stars-and-bars viewpoint for the $(5,5)$ split
  10. From a triple to a recombined block
  11. Conclusion

Key Result

Theorem 1

Let $\mathcal{B}$ be the family of $910$ blocks defined in eq:B. For every $S\subset[60]$ with $|S|=6$, there exists $B\in\mathcal{B}$ such that $|S\cap B|\ge 3$. Equivalently, $\mathcal{B}$ has covering radius $3$ in $J(60,6)$ with respect to $d_J$ in eq:johnson-distance.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Remark 1: What the bound does and does not say
  • Remark 2: Why the two-halves split works
  • Proposition 1: Generalization to $\lbrack 2m\rbrack$, $m$ even
  • proof
  • Lemma 1
  • proof
  • Proposition 2: Triple intersection under a $(5,5)$ split
  • proof