TY - CHAP A1 - Mundinger, Konrad A1 - Zimmer, Max A1 - Kiem, Aldo A1 - Spiegel, Christoph A1 - Pokutta, Sebastian T1 - Neural Discovery in Mathematics: Do Machines Dream of Colored Planes? T2 - Proceedings of the 42nd International Conference on Machine Learning Y1 - 2025 UR - https://raw.githubusercontent.com/mlresearch/v267/main/assets/mundinger25a/mundinger25a.pdf VL - 267 SP - 45236 EP - 45255 ER - TY - JOUR A1 - Parczyk, Olaf A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Szabó, Tibor T1 - New Ramsey multiplicity bounds and search heuristics JF - Foundations of Computational Mathematics Y1 - 2024 U6 - https://doi.org/10.1007/s10208-024-09675-6 ER - TY - JOUR A1 - Mundinger, Konrad A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Zimmer, Max T1 - Extending the Continuum of Six-Colorings JF - Geombinatorics Quarterly Y1 - 2024 VL - 34 IS - 1 SP - 20 EP - 29 ER - TY - CHAP A1 - Mundinger, Konrad A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Zimmer, Max T1 - Extending the Continuum of Six-Colorings T2 - Discrete Mathematics Days 2024 Y1 - 2024 U6 - https://doi.org/10.37536/TYSP5643 SP - 178 EP - 183 ER - TY - JOUR A1 - Böttcher, Julia A1 - Parczyk, Olaf A1 - Sgueglia, Amedeo A1 - Skokan, Jozef T1 - The square of a Hamilton cycle in randomly perturbed graphs JF - Random Structures & Algorithms Y1 - 2024 U6 - https://doi.org/10.1002/rsa.21215 VL - 65 IS - 2 SP - 342 EP - 386 ER - TY - CHAP A1 - Parczyk, Olaf A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Szabó, Tibor T1 - New Ramsey Multiplicity Bounds and Search Heuristics T2 - Discrete Mathematics Days N2 - We study two related problems concerning the number of monochromatic cliques in two-colorings of the complete graph that go back to questions of Erdős. Most notably, we improve the 25-year-old upper bounds of Thomason on the Ramsey multiplicity of K4 and K5 and we settle the minimum number of independent sets of size 4 in graphs with clique number at most 4. Motivated by the elusiveness of the symmetric Ramsey multiplicity problem, we also introduce an off-diagonal variant and obtain tight results when counting monochromatic K4 or K5 in only one of the colors and triangles in the other. The extremal constructions for each problem turn out to be blow-ups of a finite graph and were found through search heuristics. They are complemented by lower bounds and stability results established using Flag Algebras, resulting in a fully computer-assisted approach. More broadly, these problems lead us to the study of the region of possible pairs of clique and independent set densities that can be realized as the limit of some sequence of graphs. Y1 - 2022 ER - TY - CHAP A1 - Parczyk, Olaf A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Szabó, Tibor T1 - Fully Computer-assisted Proofs in Extremal Combinatorics T2 - Proceedings of the AAAI Conference on Artificial Intelligence N2 - We present a fully computer-assisted proof system for solving a particular family of problems in Extremal Combinatorics. Existing techniques using Flag Algebras have proven powerful in the past, but have so far lacked a computational counterpart to derive matching constructive bounds. We demonstrate that common search heuristics are capable of finding constructions far beyond the reach of human intuition. Additionally, the most obvious downside of such heuristics, namely a missing guarantee of global optimality, can often be fully eliminated in this case through lower bounds and stability results coming from the Flag Algebra approach. To illustrate the potential of this approach, we study two related and well-known problems in Extremal Graph Theory that go back to questions of Erdős from the 60s. Most notably, we present the first major improvement in the upper bound of the Ramsey multiplicity of the complete graph on 4 vertices in 25 years, precisely determine the first off-diagonal Ramsey multiplicity number, and settle the minimum number of independent sets of size four in graphs with clique number strictly less than five. Y1 - 2023 U6 - https://doi.org/10.1609/aaai.v37i10.26470 VL - 37 IS - 10 SP - 12482 EP - 12490 ER -