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The generalized trifference problem

Anurag Bishnoi, Bartłomiej Kielak, Benedek Kovács, Zoltán Lóránt Nagy, Gábor Somlai, Máté Vizer, Zeyu Zheng

TL;DR

This work generalizes the classical trifference problem to the $m$-trifferent setting over a ternary alphabet, defining $T(n,m)$ as the maximum size of a code in $\{0,1,2\}^n$ with at least $m$ triffering coordinates among every triple. It proves a phase transition near $m \approx \frac{2}{9}n$ and develops tight upper and lower bounds across regimes, including a refined bound for large $m$. The paper also studies the linear version, establishing an equivalence with $m$-minimal codes and strong blocking sets in finite geometry, and provides both upper and lower bounds for $T_L(n,m)$ along with explicit constructions and computational results. It connects these combinatorial questions to the sunflower conjecture, showing structural implications via $F(C)$ and sunflower-free families, and discusses explicit constructions via expander graphs. Overall, the results advance understanding of how $m$-trifference scales with $n$, offer practical bounds and constructions, and reveal deep links to finite geometry and extremal set theory.

Abstract

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.

The generalized trifference problem

TL;DR

This work generalizes the classical trifference problem to the -trifferent setting over a ternary alphabet, defining as the maximum size of a code in with at least triffering coordinates among every triple. It proves a phase transition near and develops tight upper and lower bounds across regimes, including a refined bound for large . The paper also studies the linear version, establishing an equivalence with -minimal codes and strong blocking sets in finite geometry, and provides both upper and lower bounds for along with explicit constructions and computational results. It connects these combinatorial questions to the sunflower conjecture, showing structural implications via and sunflower-free families, and discusses explicit constructions via expander graphs. Overall, the results advance understanding of how -trifference scales with , offer practical bounds and constructions, and reveal deep links to finite geometry and extremal set theory.

Abstract

We study the problem of finding the largest number of ternary vectors of length such that for any three distinct vectors there are at least coordinates where they pairwise differ. For , this is the classical trifference problem which is wide open. We prove upper and lower bounds on for various ranges of the parameter and determine the phase transition threshold on where jumps from constant to exponential in . By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.
Paper Structure (10 sections, 21 theorems, 53 equations, 1 figure)

This paper contains 10 sections, 21 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Proof of the lower and upper bounds
  4. The linear version
  5. The upper bound
  6. The lower bound
  7. Concluding remarks
  8. Computational results and explicit constructions
  9. Explicit constructions
  10. Relation to the sunflower conjecture

Key Result

Theorem 1.2

Suppose that $\varepsilon>0$ is a real number. If $m>(\frac{2}{9}+\varepsilon)n$, then $T(n,m)$ is bounded above by a constant depending only on $\varepsilon$. On the other hand, if $m<(\frac{2}{9}-\varepsilon)n$, then $T(n,m)$ is bounded from below by an exponential function $(1+\delta)^n$, where $

Figures (1)

  • Figure 1: Graph indicating various bounds of the form $(b(\lambda)(1\pm o(1))^n$ on $T(n,\lambda n)$

Theorems & Definitions (42)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • proof
  • Corollary 1.4
  • Theorem 1.5: Upper bound
  • Theorem 1.6
  • Theorem 1.7: Refined upper bound when $m$ is large
  • Theorem 1.8: Lower bound
  • Definition 2.1
  • ...and 32 more