## What we are interested in

Computers and Artificial Intelligence have always been an important tool for mathematicians, allowing one to gather data and extrapolate connections to formulate conjectures, exhaustively execute case analysis too large to be done by hande, solve underlying optimization problems, or to formalize and verify proofs. We are interested in exploring the many ways in which computational tools can be used to advance mathematics and formulate novel results by leveraging our unique knowledge of and access to the ZIBâ€™s computational resources.

## đź§‘â€ŤđźŽ“ Members

## đź”¬ Projects

Formal proof verification can both ensure proof correctness and provide new tools and insights to mathematicians. The goals of this project include creating resources for students and researchers, verifying relevant results, improving proof tactics, and exploring Machine Learning approaches.

This project aims to obtain new bounds in Extremal Combinatorics through an application of flag algebras. The goal is to both improve the underlying computational aspects for existing problems as well as to further develop the theory of flag algebras to extend it to new areas of application.

Extremal Combinatorics focuses on the maximum or minimum sizes of discrete structures with specific properties, posing significant challenges due to their complexity. Traditional computational approaches often fail due to exponential growth in search spaces, but recent AI advancements, especially in Reinforcement Learning, offer new potential. Applying these AI methods could provide insights into combinatorial problems while also enhancing the understanding of AI techniques in complex, sparse reward environments.

## đź’¬ Talks and posters

### Conference and workshop talks

- Oct 2024
*Extending the Continuum of Six-Colorings*by Christoph Spiegel

41st KolKom Conference, Heidelberg [PDF]- Jul 2024
*Extending the Continuum of Six-Colorings*by Christoph Spiegel

13th DMD Conference, AlcalĂˇ de Henares [PDF]- Jul 2024
*The Four-Color Ramsey Multiplicity of Triangles*by Aldo Kiem

25th ISMP Conference, MontrĂ©al- Jul 2024
*The Four-Color Ramsey Multiplicity of Triangles*by Aldo Kiem

13th DMD Conference, AlcalĂˇ de Henares- Jun 2024
*On the Maximum Diameter of D-dimensional Simplicial Complexes*by Silas Rathke

Mittagsseminar Discrete Mathematics, Berlin

## View More / Less

- May 2024
*On the Maximum Diameter of D-dimensional Simplicial Complexes*by Silas Rathke

Berlin-PoznaĹ„ Seminar, Berlin- Dec 2023
*New Ramsey Multiplicity Bounds and Search Heuristics*by Tibor SzabĂł

45th Australasian Combinatorics Conference, Perth- Aug 2023
*The Rado Multiplicity Problem in Vector Spaces Over Finite Fields*by Christoph Spiegel

12th EuroComb Conference, Prague [PDF]- Aug 2023
*Computational Challenges in Flag Algebra Proofs*by Christoph Spiegel

ICIAM 2023 Minisymposium: Advances in Optimization I, Tokyo [PDF]- Aug 2023
*Flag Algebras in Additive Combinatorics*by Christoph Spiegel

5th DOxML Conference, Tokyo [PDF]- Jun 2023
*New Ramsey Multiplicity Bounds and Search Heuristics*by Olaf Parczyk

FoCM 2023 Workshop I.3 Workshop, Paris- Jun 2023
*Towards Flag Algebras in Additive Combinatorics*by Christoph Spiegel

FoCM 2023 Workshop I.3 Workshop, Paris [PDF]- Jun 2023
*The 4-color Ramsey Multiplicity of Triangles*by Aldo Kiem

FoCM 2023 Workshop I.3 Workshop, Paris- May 2023
*Towards Flag Algebras in Additive Combinatorics*by Christoph Spiegel

15th CANT Conference, New York [PDF]- May 2023
*New Ramsey Multiplicity Bounds and Search Heuristics*by Tibor SzabĂł

Berlin-PoznaĹ„ Seminar, Hamburg- Mar 2023
*Computer-assisted Proofs in Extremal Combinatorics*by Christoph Spiegel

Optimization and ML Workshop, Waischenfeld [PDF]- Feb 2023
*Fully Computer-assisted Proofs in Extremal Combinatorics*by Christoph Spiegel

37th AAAI Conference, Washington, DC [PDF]- Jan 2023
*Fully Computer-assisted Proof in Extremal Combinatorics*by Christoph Spiegel

Combinatorial Optimization Workshop, Aussois [PDF]- Dec 2022
*Leveraging Combinatorial Symmetries in Flag Algebra-based SDP Formulations*by Christoph Spiegel

Recent Advances in Optimization Workshop, Toronto [PDF]- Sep 2022
*Proofs in Extremal Combinatorics Through Optimization*by Christoph Spiegel

6th RIKEN-MODAL Workshop, Tokyo / Fukuoka [PDF]- Sep 2022
*New Ramsey Multiplicity Bounds and Search Heuristics*by Olaf Parczyk

DMV Annual Meeting, Berlin- Aug 2022
*New Ramsey Multiplicity Bounds and Search Heuristics*by Olaf Parczyk

20th RS&A Conference, Gniezno- Jul 2022
*New Ramsey Multiplicity Bounds and Search Heuristics*by Christoph Spiegel

12th DMD Conference, Santander [PDF]- Jun 2022
*New Ramsey Multiplicity Bounds and Search Heuristics*by Olaf Parczyk

Third Southwestern German Workshop on Graph Theory, Heidelberg- Jun 2019
*Odd Cycle Games and Connected Rules*by Christoph Spiegel

36th Postgraduate Combinatorial Conference, Oxford [PDF]- Jun 2018
*On a Problem of SĂˇrkĂ¶zy and SĂłs for Multivariate Linear Forms*by Christoph Spiegel

11th DMD Conference, Sevilla [PDF]- May 2018
*Going Beyond 2.4 in Freiman's 2.4k-theorem*by Christoph Spiegel

10th CANT Conference, New York [PDF]- Sep 2017
*Rado Positional Games*by Christoph Spiegel

The Music of Numbers Conference, Madrid [PDF]- Jun 2017
*Generalized Positional Van Der Waerden Games*by Christoph Spiegel

Interactions with Combinatorics [PDF]

### Research seminar talks

- Jun 2024
*Categorification of Flag Algebras*by Aldo Kiem

LIMDA Seminar, Barcelona- May 2024
*New Ramsey Multiplicity Bounds and Search Heuristics*by Tibor SzabĂł

Research Seminar in Discrete Mathematics [video]- Mar 2024
*New Ramsey Multiplicity Bounds and Search Heuristics*by Tibor SzabĂł

Atlanta Lecture Series Combinatorics and Graph Theory XXVIII, Atlanta- Feb 2024
*Maximal Diameter of Simplicial D-complexes*by Olaf Parczyk

Seminar on Combinatorics, Games and Optimisation, London- Nov 2023
*New Ramsey Multiplicity Bounds and Search Heuristics*by Olaf Parczyk

Seminar on Discrete Mathematics at Adam Mickiewicz University, Poznan

## View More / Less

- Feb 2023
*The Four-color Ramsey Multiplicity of Triangles*by Christoph Spiegel

LIMDA Seminar, Barcelona [PDF]- Nov 2022
*Flag Algebras for Edge-colored Graphs*by Aldo Kiem

Research Seminar Combinatorics, Berlin- Apr 2022
*New Ramsey Multiplicity Bounds and Search Heuristics*by Christoph Spiegel

LIMDA Seminar, Barcelona- Nov 2019
*A Step Beyond Freiman's Theorem for Set Addition Modulo a Prime*by Christoph Spiegel

Research Seminar Combinatorics, Berlin- Nov 2019
*The Probabilistic Intuition in Positional Games*by Christoph Spiegel

Research Seminar Large Networks and Random Graphs, Ilmenau- May 2019
*Intervals in the Hales-Jewett Theorem*by Christoph Spiegel

Seminar on Combinatorics, Games and Optimisation, London- May 2019
*Intervals in the Hales-Jewett Theorem*by Christoph Spiegel

Combinatorial Theory Seminar, Oxford- Dec 2018
*Intervals in the Hales-Jewett Theorem*by Christoph Spiegel

Research Seminar Combinatorics, Berlin- Mar 2018
*Going Beyond 2.4 in Freiman's 2.4k-theorem*by Christoph Spiegel

GRAPHS at IMPA, Rio de Janeiro- Dec 2017
*On a Question of SĂˇrkĂ¶zy and SĂłs*by Christoph Spiegel

Research Seminar Combinatorics, Berlin- Oct 2017
*Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems*by Christoph Spiegel

LIMDA Seminar, Barcelona- May 2017
*Random Strategies Are Nearly Optimal for Generalized Van Der Waerden Games*by Christoph Spiegel

LIMDA Seminar, Barcelona- Mar 2016
*Threshold Functions for Systems of Equations in Random Sets*by Christoph Spiegel

LIMDA Seminar, Barcelona- Feb 2016
*Van Der Waerden Games (II)*by Christoph Spiegel

Research Seminar Combinatorics, Berlin- Jan 2016
*Van Der Waerden Games (I)*by Christoph Spiegel

Research Seminar Combinatorics, Berlin- Oct 2015
*Threshold Functions for Systems of Equations in Random Sets*by Christoph Spiegel

Research Seminar Combinatorics, Berlin

### Poster presentations

- Jul 2024
*Categorification of Flag Algebras*by Aldo Kiem

13th DMD Conference, AlcalĂˇ de Henares

## đź“ť Publications and preprints

- Parczyk, O., Pokutta, S., Spiegel, C., and SzabĂł, T. (2024). New Ramsey Multiplicity Bounds and Search Heuristics.
*Foundations of Computational Mathematics*. DOI: 10.1007/s10208-024-09675-6 [arXiv] [code]## [BibTeX]

- Kiem, A., Pokutta, S., and Spiegel, C. (2024). The 4-color Ramsey Multiplicity of Triangles.
*Proceedings of Discrete Mathematics Days*. [URL] [arXiv] [code]## [BibTeX]

- Kiem, A., Pokutta, S., and Spiegel, C. (2024). Categorification of Flag Algebras.
*Proceedings of Discrete Mathematics Days*. [URL]## [BibTeX]

- Mundinger, K., Pokutta, S., Spiegel, C., and Zimmer, M. (2024). Extending the Continuum of Six-Colorings.
*Geombinatorics Quarterly*,*XXXIV*. [URL] [arXiv]## [BibTeX]

- Mundinger, K., Pokutta, S., Spiegel, C., and Zimmer, M. (2024). Extending the Continuum of Six-Colorings.
*Proceedings of Discrete Mathematics Days*. [URL] [arXiv]## [BibTeX]

- Parczyk, O., and Spiegel, C. (2024).
*An Unsure Note on an Un-Schur Problem*. [arXiv]## [BibTeX]

- Parczyk, O., Pokutta, S., Spiegel, C., and SzabĂł, T. (2023). Fully Computer-assisted Proofs in Extremal Combinatorics.
*Proceedings of AAAI Conference on Artificial Intelligence*. DOI: 10.1609/aaai.v37i10.26470 [URL] [arXiv] [code]## [BibTeX]

- RuĂ©, J. J., and Spiegel, C. (2023). The Rado Multiplicity Problem in Vector Spaces Over Finite Fields.
*Proceedings of European Conference on Combinatorics*. DOI: 10.5817/CZ.MUNI.EUROCOMB23-108 [URL] [arXiv] [code]## [BibTeX]

- KamÄŤev, N., and Spiegel, C. (2022). Another Note on Intervals in the Hales-Jewett Theorem.
*Electronic Journal of Combinatorics*,*29*(1). DOI: 10.37236/9400 [URL] [arXiv]## [BibTeX]

- Braun, G., Pokutta, S., and Weismantel, R. (2022).
*Alternating Linear Minimization: Revisiting von Neumannâ€™s Alternating Projections*. [arXiv] [slides] [video]## [BibTeX]

- Parczyk, O., Pokutta, S., Spiegel, C., and SzabĂł, T. (2022). New Ramsey Multiplicity Bounds and Search Heuristics.
*Proceedings of Discrete Mathematics Days*. [arXiv] [code]## [BibTeX]

- Cao-Labora, G., RuĂ©, J. J., and Spiegel, C. (2021-10). An ErdĹ‘s-Fuchs Theorem for Ordered Representation Functions.
*Ramanujan Journal*,*56*, 183â€“2091. DOI: 10.1007/s11139-020-00326-2 [URL] [arXiv]## [BibTeX]

- Fabian, D., RuĂ©, J. J., and Spiegel, C. (2021-08). On Strong Infinite Sidon and Bâ‚• Sets and Random Sets of Integers.
*Journal of Combinatorial Theory, Series A*,*182*. DOI: 10.1016/j.jcta.2021.105460 [URL] [arXiv]## [BibTeX]

- Corsten, J., Mond, A., Pokrovskiy, A., Spiegel, C., and SzabĂł, T. (2020-10). On the Odd Cycle Game and Connected Rules.
*European Journal of Combinatorics*,*89*. DOI: 10.1016/j.ejc.2020.103140 [URL] [arXiv]## [BibTeX]

- RuĂ©, J. J., and Spiegel, C. (2020). On a Problem of SĂˇrkĂ¶zy and SĂłs for Multivariate Linear Forms.
*Revista MatemĂˇtica Iberoamericana*,*36*(7), 2107â€“2119. DOI: 10.4171/RMI/1193 [URL] [arXiv]## [BibTeX]

- Candela, P., Serra, O., and Spiegel, C. (2020). A Step Beyond FreÄmanâ€™s Theorem for Set Addition Modulo a Prime.
*Journal De ThĂ©orie Des Nombres De Bordeaux*,*32*(1), 275â€“289. DOI: 10.5802/jtnb.1122 [URL] [arXiv]## [BibTeX]

- Kusch, C., RuĂ©, J. J., Spiegel, C., and SzabĂł, T. (2019-09). On the Optimality of the Uniform Random Strategy.
*Random Structures & Algorithms*,*55*(2), 371â€“401. DOI: 10.1002/rsa.20829 [URL] [arXiv]## [BibTeX]

- FreÄman, G. A., Serra, O., and Spiegel, C. (2019). Additive Volume of Sets Contained in Few Arithmetic Progressions.
*INTEGERS*,*19*. [URL] [arXiv]## [BibTeX]

- Salia, N., Spiegel, C., Tompkins, C., and Zamora, O. (2019).
*Independent Chains in Acyclic Posets*. [arXiv]## [BibTeX]

- RuĂ©, J. J., Spiegel, C., and ZumalacĂˇrregui, A. (2018). Threshold Functions and Poisson Convergence for Systems of Equations in Random Sets.
*Mathematische Zeitschrift*,*288*, 333â€“360. DOI: 10.1007/s00209-017-1891-2 [URL] [arXiv]## [BibTeX]

- RuĂ©, J. J., and Spiegel, C. (2018). On a Problem of SĂˇrkĂ¶zy and SĂłs for Multivariate Linear Forms.
*Proceedings of Discrete Mathematics Days*. [URL] [arXiv]## [BibTeX]

- Kusch, C., RuĂ©, J. J., Spiegel, C., and SzabĂł, T. (2017). Random Strategies Are Nearly Optimal for Generalized Van Der Waerden Games.
*Proceedings of European Conference on Combinatorics*. [URL] [arXiv]## [BibTeX]

- Spiegel, C. (2017). A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems.
*Electronic Journal of Combinatorics*,*24*(3). DOI: 10.37236/6730 [URL] [arXiv]## [BibTeX]

- BĂ¶ttcher, J., Frankl, N., Cecchelli, D. M., Skokan, J., and Parczyk, O.
*Graphs with Large Minimum Degree and No Small Odd Cycles Are 3-colourable*. [arXiv]## [BibTeX]

- Mattos, L., Cecchelli, D. M., and Parczyk, O.
*On Product Schur Triples in the Integers*. [arXiv]## [BibTeX]

- DĂaz, A. E., Gupta, P., Cecchelli, D. M., Parczyk, O., and Sgueglia, A.
*Diracâ€™s Theorem for Graphs of Bounded Bandwidth*. [arXiv]## [BibTeX]

- Illingworth, F., Lang, R., MĂĽyesser, A., Parczyk, O., and Sgueglia, A.
*Spanning Spheres in Dirac Hypergraphs*. [arXiv]## [BibTeX]