Work in the first work package was done mainly by secondments to Princeton University. During the secondments, we worked mainly with Maria Chudnovsky (Princeton University), Paul Seymour (Princeton University), Noga Alon (Princeton University) Sophie Spirkl and Cemil Dibek, a current doctoral student of Maria Chudnovsky. Ron Aharoni in collaboration with other authors proved lower bounds on the matching number of certain hypergraphs by using a topological version of Hall's theorem. These bounds yield rainbow versions of the KKM theorem for products of simplices, which in turn are used to obtain some results on multiple-cake division, and on rainbow matchings in families of d-intervals. Furthermore, CoSP has interesting results on rainbow structures. Furthermore, Ron Holzman improved the previously known results of generalizations of Erdős–Spencer theorem.
Work in the second package was done by secondments to Rutgers University and to Simon-Fraser University. We worked mainly with Michael Saks of Rutgers University and with Bojan Mohar, Matt DeVos and David Wood from Simon Fraser University. We addressed the cryptographic problem of designing oblivious RAM with partial progress in understanding the problems. Another cutting edge research studied a close connection between circuit complexity of sorting and a well-known conjecture on network coding. We explored a new research direction: algorithms, optimization and dynamics in the Algorithmic Game Theory. This is an exciting attractive and robust area to continue working on.
Work in the third work package was done by secondments at Simon Fraser University and at Rutgers University. We were working mostly with Bojan Mohar, Matt DeVos and Pavol Hell and with Gregory Cherlin of Rutgers University. Within this research package, we worked on a counting problem for group connectivity. Furthermore, Jan Bok in collaboration with prof. Hell, Jedličková and other collaborators obtained a partial dichotomy for the list homomorphism problem for signed graphs.
Work package 4 –Training. Each year, CoSP organized a summer research experience and training for young talented researchers. Sixteeen to twenty participants per year conducted research on current problems in collaboration with top-grade researchers from the USA. For some participants, CoSP provided the first opportunity to publish a research paper and motivated them to pursue academic career. The training included also technical lectures and soft-skills training (scientific writing, ethics, research skills…). CoSP trained in total 76 talented young researchers between 2019 and 2024.
Events organized by CoSP: CoSP School on algorithms (Rutgers University, USA, 2019), CoSP School on topological methods (Charles University, CZ, 2019), CoSP School on homomorphisms (Charles University, CZ, 2019), CoSP school+workshop on matchings (Charles University, CZ, 2022), CoSP workshop on algorithms (Rutgers University, USA, 2024), CoSP ZOOM seminar on topological combinatorics, yearly CoSP student workshop (Charles University, CZ), CoSP annual, midterm and final meetings.
