Session I.3 - Graph Theory and Combinatorics
Tuesday, June 13, 17:00 ~ 17:30
Towards flag algebras in additive combinatorics
Christoph Spiegel
Zuse Institute Berlin, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic $3$-APs in 3-colorings of $\mathbb{F}_3^n$ as well as obtaining the first non-trivial lower bound for the minimum number of monochromatic $4$-APs in 2-colorings of $\mathbb{F}_5^n$. The former parallels results by Cumings et al (2013) in extremal graph theory and the latter improves upon results of Saad and Wolf (2017) The lower bounds are notably obtained by extending the flag algebra calculus of Razborov (2007) to additive structures in vector spaces over finite fields.
This work is available on arXiv: https://arxiv.org/abs/2304.00400
Joint work with Juanjo Rué (Universitat Politècnica de Catalunya and Centre de Recerca Matemàtica, Spain).