# Schur Number Five

@inproceedings{Heule2018SchurNF, title={Schur Number Five}, author={Marijn J. H. Heule}, booktitle={AAAI}, year={2018} }

We present the solution of a century-old problem known as Schur Number Five: What is the largest (natural) number $n$ such that there exists a five-coloring of the positive numbers up to $n$ without a monochromatic solution of the equation $a + b = c$? We obtained the solution, $n = 160$, by encoding the problem into propositional logic and applying massively parallel satisfiability solving techniques on the resulting formula. We constructed and validated a proof of the solution to increase…

## 45 Citations

A SAT-based Resolution of Lam's Problem

- Computer ScienceAAAI
- 2021

This work uncovered consistency issues in both previous searches, highlighting the difficulty of relying on special-purpose search code for nonexistence results and using satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party.

The SAT+CAS paradigm and the Williamson conjecture

- Mathematics, Computer ScienceACCA
- 2019

The Williamson conjecture from combinatorial design theory is studied and all Williamson matrices of orders divisible by 2 or 3 up to and including 70 are completely enumerated.

Schur numbers involving rainbow colorings

- MathematicsArs Math. Contemp.
- 2020

Two different generalizations of Schur numbers that involve rainbow colorings are introduced, and it is shown that for all n, the Gallai-Schur number G S ( n) is the least natural number such that every n -coloring that lacks rainbow solutions to the equation a + b = c necessarily contains a monochromatic solution to this equation.

The SAT+CAS method for combinatorial search with applications to best matrices

- Mathematics, Computer ScienceAnnals of Mathematics and Artificial Intelligence
- 2019

An overview of the SAT+CAS method that combines satisfiability checkers (SAT solvers) and computer algebra systems (CAS) to resolve combinatorial conjectures, and present new results vis-à-vis best matrices.

A SAT+CAS Approach to Finding Good Matrices: New Examples and Counterexamples

- MathematicsAAAI
- 2019

The method applies the SAT+CAS paradigm of combining computer algebra functionality with modern SAT solvers to efficiently search large spaces which are specified by both algebraic and logical constraints.

A nonexistence certificate for projective planes of order ten with weight 15 codewords

- Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 2020

This work shows how the performance of the SAT solver can be dramatically increased by employing functionality from a computer algebra system (CAS) and runs significantly faster than all other published searches verifying this result.

Strong Extension-Free Proof Systems

- Computer ScienceJournal of Automated Reasoning
- 2019

This work introduces proof systems for propositional logic that admit short proofs of hard formulas as well as the succinct expression of most techniques used by modern SAT solvers, and guarantees that these added clauses are redundant.

Applying Computer Algebra Systems and SAT Solvers to the Williamson Conjecture

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2020

Finding the Hardest Formulas for Resolution

- MathematicsCP
- 2020

The first ten resolution hardness numbers are computed by a candidate filtering and symmetry breaking search scheme for limiting the number of potential candidates for formulas and an efficient SAT encoding for computing a shortest resolution proof of a given candidate formula.

N ov 2 01 8 A SAT + CAS Approach to Finding Good Matrices : New Examples and Counterexamples Curtis Bright

- Mathematics
- 2018

We enumerate all circulant good matrices with odd orders divisible by 3 up to order 70. As a consequence of this we find a previously overlooked set of good matrices of order 27 and a new set of good…

## References

SHOWING 1-10 OF 46 REFERENCES

Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

- MathematicsSAT
- 2016

This work solves the boolean Pythagorean Triples problem by using the Cube-and-Conquer paradigm, a hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers, and produced and verified a proof in the DRAT format, which is almost 200 terabytes in size.

Weak Schur numbers and the search for G.W. Walker's lost partitions

- MathematicsComput. Math. Appl.
- 2012

Backdoors To Typical Case Complexity

- Computer ScienceIJCAI
- 2003

This work proposes a new framework for studying the complexity of reasoning and constraint processing methods, which incorporates general structural properties observed in practical problem instances into the formal complexity analysis and introduces a notion of "backdoors", which are small sets of variables that capture the overall combinatorics of the problem instance.

Efficient Certified Resolution Proof Checking

- Computer Science, MathematicsTACAS
- 2017

A novel propositional proof tracing format that eliminates complex processing, thus enabling efficient (formal) proof checking, and formally verify the recent 200 TB proof of the Boolean Pythagorean Triples conjecture.

A backbone-search heuristic for efficient solving of hard 3-SAT formulae

- MathematicsIJCAI
- 2001

A heuristic search for variables belonging to the backbone of a 3-SAT formula which are chosen as branch nodes for the tree developed by a DPL-type procedure is defined, making it possible to handle unsatisfiable hard 3- SAT formulae up to 700 variables.

Cube and Conquer: Guiding CDCL SAT Solvers by Lookaheads

- Computer ScienceHaifa Verification Conference
- 2011

This work presents a new approach, called cube-and-conquer, targeted at reducing solving time on hard instances, and finds that this hybrid approach outperforms both lookahead and conflict-driven solvers.

The van der Waerden Number W(2, 6) Is 1132

- Computer ScienceExp. Math.
- 2008

The exhaustive search showing that W(2, 6) = 1132 was carried out by formulating the problem as a satisfiability (SAT) question for a Boolean formula in conjunctive normal form (CNF) and then using a SAT solver specifically designed for the problem.

Inprocessing Rules

- Computer ScienceIJCAR
- 2012

The formal underpinnings of inprocessing SAT solving are established via an abstract inprocessing framework that covers a wide range of modern SAT solving techniques.

Expressing Symmetry Breaking in DRAT Proofs

- Computer ScienceCADE
- 2015

This work presents a method to express the addition of symmetry-breaking predicates in DRAT, a clausal proof format supported by top-tier solvers and validated these proofs with an ACL2-based, mechanically-verified DRAT proof checker and the proof-checking tool of SAT Competition 2014.