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PatternBoost: Constructions in Mathematics with a Little Help from AI

François Charton, Jordan S. Ellenberg, Adam Zsolt Wagner, Geordie Williamson

TL;DR

PatternBoost introduces a practical two-phase framework that alternates local greedy construction with transformer-guided global search to discover new mathematical constructions. Demonstrated across multiple extremal combinatorics problems, the approach yields state-of-the-art or near-state-of-the-art results, including counterexamples to long-standing conjectures and improvements on several bounds. The work highlights the method’s versatility, the importance of model capacity and data representation, and its potential to augment, rather than replace, human insight in mathematical discovery. By providing accessible tooling and a generalizable workflow, PatternBoost could become a valuable partner for researchers exploring complex combinatorial landscapes.

Abstract

We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first ``local'' phase, a classical search algorithm is used to produce many desirable constructions. In the second ``global'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years.

PatternBoost: Constructions in Mathematics with a Little Help from AI

TL;DR

PatternBoost introduces a practical two-phase framework that alternates local greedy construction with transformer-guided global search to discover new mathematical constructions. Demonstrated across multiple extremal combinatorics problems, the approach yields state-of-the-art or near-state-of-the-art results, including counterexamples to long-standing conjectures and improvements on several bounds. The work highlights the method’s versatility, the importance of model capacity and data representation, and its potential to augment, rather than replace, human insight in mathematical discovery. By providing accessible tooling and a generalizable workflow, PatternBoost could become a valuable partner for researchers exploring complex combinatorial landscapes.

Abstract

We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first ``local'' phase, a classical search algorithm is used to produce many desirable constructions. In the second ``global'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years.

Paper Structure

This paper contains 24 sections, 2 theorems, 11 equations, 19 figures, 5 tables.

Table of Contents

  1. Introduction
  2. A simple illustrative problem: many edges with no triangles
  3. Hard and easy problems
  4. The no-squares problem
  5. A first experiment
  6. The advantage of larger models, and better tokenizers
  7. Scaling PatternBoost
  8. The 312-avoiding permanent problem
  9. Combining methods
  10. Spanning subgraphs of the $d$-cube with diameter $d$
  11. Other problems
  12. No isosceles triangles
  13. The "no 5 on a sphere" problem
  14. Saturated Sperner systems of small size
  15. Cross-Sperner systems
  16. ...and 9 more sections

Key Result

Theorem 4.2

Let $d\geq 5$. There is a collection of $\frac{41}{32}2^d\approx 1.28\cdot 2^d$ proper sub-boxes that cover every point of $\{0,1,2\}^d$ exactly twice.

Figures (19)

  • Figure 1: Left: score distribution of all the examples we generated by greedily adding edges for as long as possible. Right: the score distribution of the best 25% of the examples from the left, which we will use as a training set.
  • Figure 2: Flattening the adjacency matrix
  • Figure 3: The score distribution in the second generation. The peak has shifted from 66 to 70.
  • Figure 4: The evolution of scores across the next 5 generations. By the end, the model has learned to generate many different complete bipartite graphs.
  • Figure 5: The results of adding random edges to empty graph for as long as possible, without creating any 4-cycles, 50M times in total.
  • ...and 14 more figures

Theorems & Definitions (3)

  • Theorem 4.2
  • Proposition A.1
  • proof