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Term Coding for Extremal Combinatorics: Dispersion and Complexity Dichotomies

Søren Riis

TL;DR

Term Coding provides a quantifier-free, negation-free framework to encode extremal combinatorial problems as finite term-equation systems, optionally with non-equality constraints. The authors relate the maximum solution size $\max_{\mathcal{I}}(\Gamma,n)$ to graph guessing numbers and entropy, and extend the framework to multi-sorted settings, enabling designs, finite geometries, and mixed coding scenarios to be analyzed uniformly. A main result is a finite-bounds theorem tying code size to a guessing-number of an associated graph; dispersion is studied as a natural subclass with a striking complexity dichotomy in the single-sorted case: deciding if the maximum can reach $n^r$ is undecidable, while deciding if it eventually exceeds $n^r+1$ is polynomial-time decidable. The work also generalises to FO finite satisfiability via multi-sorted Term Coding, showing undecidability in the general setting and establishing a robust toolkit (normalisation, diversification, guessing-number bounds) for extremal and existence questions. Altogether, Term Coding unifies and extends extremal combinatorics, finite model theory, and network coding, revealing deep connections between algebraic representations and information-theoretic limits with potential implications for combinatorial design and algorithmic reasoning.

Abstract

We introduce \emph{Term Coding}, a novel framework for analysing extremal problems in discrete mathematics by encoding them as finite systems of \emph{term equations} (and, optionally, \emph{non-equality constraints}). In its basic form, all variables range over a single domain, and we seek an interpretation of the function symbols that \emph{maximises} the number of solutions to these constraints. This perspective unifies classical questions in extremal combinatorics, network/index coding, and finite model theory. We further develop \emph{multi-sorted Term Coding}, a more general approach in which variables may be of different sorts (e.g., points, lines, blocks, colours, labels), possibly supplemented by variable-inequality constraints to enforce distinctness. This extension captures sophisticated structures such as block designs, finite geometries, and mixed coding scenarios within a single logical formalism. Our main result shows how to determine (up to a constant) the maximum number of solutions \(\max_{\mathcal{I}}(Γ,n)\) for any system of term equations (possibly including non-equality constraints) by relating it to \emph{graph guessing numbers} and \emph{entropy measures}. Finally, we focus on \emph{dispersion problems}, an expressive subclass of these constraints. We discover a striking complexity dichotomy: deciding whether, for a given integer \(r\), the maximum code size that reaches \(n^{r}\) is \emph{undecidable}, while deciding whether it exceeds \(n^{r}\) is \emph{polynomial-time decidable}.

Term Coding for Extremal Combinatorics: Dispersion and Complexity Dichotomies

TL;DR

Term Coding provides a quantifier-free, negation-free framework to encode extremal combinatorial problems as finite term-equation systems, optionally with non-equality constraints. The authors relate the maximum solution size to graph guessing numbers and entropy, and extend the framework to multi-sorted settings, enabling designs, finite geometries, and mixed coding scenarios to be analyzed uniformly. A main result is a finite-bounds theorem tying code size to a guessing-number of an associated graph; dispersion is studied as a natural subclass with a striking complexity dichotomy in the single-sorted case: deciding if the maximum can reach is undecidable, while deciding if it eventually exceeds is polynomial-time decidable. The work also generalises to FO finite satisfiability via multi-sorted Term Coding, showing undecidability in the general setting and establishing a robust toolkit (normalisation, diversification, guessing-number bounds) for extremal and existence questions. Altogether, Term Coding unifies and extends extremal combinatorics, finite model theory, and network coding, revealing deep connections between algebraic representations and information-theoretic limits with potential implications for combinatorial design and algorithmic reasoning.

Abstract

We introduce \emph{Term Coding}, a novel framework for analysing extremal problems in discrete mathematics by encoding them as finite systems of \emph{term equations} (and, optionally, \emph{non-equality constraints}). In its basic form, all variables range over a single domain, and we seek an interpretation of the function symbols that \emph{maximises} the number of solutions to these constraints. This perspective unifies classical questions in extremal combinatorics, network/index coding, and finite model theory. We further develop \emph{multi-sorted Term Coding}, a more general approach in which variables may be of different sorts (e.g., points, lines, blocks, colours, labels), possibly supplemented by variable-inequality constraints to enforce distinctness. This extension captures sophisticated structures such as block designs, finite geometries, and mixed coding scenarios within a single logical formalism. Our main result shows how to determine (up to a constant) the maximum number of solutions \(\max_{\mathcal{I}}(Γ,n)\) for any system of term equations (possibly including non-equality constraints) by relating it to \emph{graph guessing numbers} and \emph{entropy measures}. Finally, we focus on \emph{dispersion problems}, an expressive subclass of these constraints. We discover a striking complexity dichotomy: deciding whether, for a given integer , the maximum code size that reaches is \emph{undecidable}, while deciding whether it exceeds is \emph{polynomial-time decidable}.

Paper Structure

This paper contains 61 sections, 16 theorems, 115 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. General Motivation
  3. Motivating Example: Steiner Triple Systems (t=2, k=3)
  4. Two witnesses: ideal vs. near‑miss.
  5. Term Coding: Purely Equational Single-Sorted Case
  6. Illustrative Examples
  7. An Unsolvable Variant
  8. Investigating Non-Solvability
  9. Diversification of Function Symbols
  10. Graph Guessing Game Reformulation and Our New Method
  11. Labelling of Nodes and Distinctness Constraints
  12. Deterministic Strategies and Winning Configurations.
  13. Maximal Solutions vs. Correct Guesses
  14. Computing the Guessing Number
  15. Multi-Sorted Term Coding and Non-Equality Constraints
  16. ...and 46 more sections

Key Result

Proposition 2.3

Let $\langle \Gamma, \Delta \rangle$ be any system (single‑ or multi‑sorted). Let $\langle \Gamma', \Delta' \rangle$ be obtained by flattening every compound subterm and by replacing each disequality $s\neq t$ with the atomic $x_i\neq x_j$ that references the newly introduced variables (Definition d

Figures (3)

  • Figure 1: Directed graph showing the functional dependencies for the normalised and diversified term equations derived from the unsolvable self-decoding Latin square variant (introduced in Section \ref{['sec:variant']}). The graph has six nodes, labelled according to the variables $\{x,x,y,y,z,w\}$ on the right-hand side of these six equations. An edge $u \to v$ indicates that variable $v$ depends functionally on variable $u$ in the corresponding equation. The next section analyses this dependency structure using guessing number/entropy techniques.
  • Figure 2: Variable dependency graph $G_\Gamma$ for the normalised system derived from $f(x,f(x,y))=y$ and $f(f(x,y),y)=x$, where $z=f(x,y)$.
  • Figure 3: Variable dependency graph $G_\Gamma$: the core bidirected cycle $C_5$ on $\{x,y,z,\alpha,\beta\}$. To keep the figure visually simple, we omit the auxiliary outputs (e.g., $\gamma=f(x,y)$ and $\delta=f(y,x)$) and the disequality constraint $\gamma\neq\delta$.

Theorems & Definitions (47)

  • Remark 1.1: Normalisation choices
  • Definition 2.1: Normalised disequality
  • Definition 2.2: Normalised Equation
  • Proposition 2.3: Normalisation preserves solutions for pairs
  • proof
  • Remark 2.1: Connection to Diversification
  • Proposition 2.4
  • proof
  • Definition 3.1: Variable Dependency Graph
  • Definition 3.2: Interpretation
  • ...and 37 more