## Participant Testimonials

Attending the workshop 'Geometric and topological graph theory (13w5091)' turned out as an excellent opportunity for scientific collaboration, as we managed both to initiate new collaborations as well as deepen the existing ones. In the latter regard, we had several exchanges on possible projects that could be submitted to various grant applications, as well as on completing the existing projects that are close to termination. Regarding the new collaborations, we discussed several open problems, and tested several approaches to them. With Radoslav Fulek, we investigated the gap of the crossing number additivity over cuts, with Jernej Rus, we devised a roadmap that will lead to an efficient algorithm for counting stable traces in graphs, and with Dan Archdeacon and Luis Goddyn we investigated approaches to formulating the thrackle existence problem as a stack of constraint satisfaction problems that may lead to a new algorithm for recognizing thrackle graphs. Besides the actual work, there were several interesting talks presenting deep, although sometimes counterintuitive new results.

This was an excellent workshop. I gave a talk on my most recent research project. I get several helpful comments, and made further progress on it. I also heard some wonderful talks by Kostochka and others on progress on some old problems on graph coloring, with very impressive results. I met a number of students (I knew almost all of the postdocs and faculty).

Firstly, I would like to thank all of the workshop participants and especially the organizers and Banff Centre staff who helped make the workshop productive and successful. Paul Seymour and I presented the following conjectures by Kalai and Meshulam: (1) If a graph has no induced cycles of length a multiple of 3, then the number of even independent sets minus the number of odd independent sets is at most one in absolute value (2) If a graph has no induced cycles of length a multiple of 3, then the graph has bounded chromatic number (and, as a stronger conjecture, is also 3-colorable) (3) If a graph has all of its induced subgraphs satisfying number of even independent sets minus number of odd independent sets is at most 1 in absolute value, then the graph has bounded chromatic number These conjectures generated much discussion, especially with Zhentao Li. Zhentao Li, Paul Seymour, and others, including myself, worked on adapting Matthew Plumettaz's results for graphs with no cycles of length a multiple of 3 to the dual problem of having no bonds with a multiple of 3 edges. We discussed how to prove that there is no simple, 3-edge-connected graph with no bonds having a multiple of 3 edges. Additionally, many of the presentations not only presented interesting problems to consider, they also provided possible approaches for proving the Kalai and Meshulam conjectures in whole or in part. I was able to meet with fellow workshop participants, who were friendly and eager to discuss graph theory both during the scheduled activities and outside of those activities; I look forward to having future discussions with them. The workshop was a very educational experience, one that I hope is a starting point for my future mathematical endeavors.

It was a great meeting. I learned not only excellent results, but whole new topics to study and use. One example is the talk of Jacob Fox, where I learned the topic of Turan-type problems for permutations, apart from his outstanding result solving (disproving) several old conjectures. Also I started a new project with Hehui Wu at the conference.

My participation in the BIRS workshop was excellent stimulus to my research. There were many exceptional high quality talks from Jacob Fox's talk about the new and exciting interval minors to Paul Seymour's talk on variants of Woodall's conjecture. Yet there were also many informal but equally enlightening conversations. In particular, I was able to discuss ongoing work with my colleague Zdenek Dvorak, to work on open problems with Sergey Norin, and to discuss career plans with many of the graduate students attending. Groups of us would sit around at night and tackle interesting open problems. While we were able to resolve a number of questions via our combined manpower, there were still a number left unresolved that caught our attention, for example Aharoni's conjecture that chromatic number of the union of two chordal graphs is at most the sum of the clique numbers. I am very pleased to have been so inspired.

It was a great workshop, there were many exciting talks, I would like to point out two of them. Jacob Fox's talk (Stanley-Wilf limits are typically exponential) was extremely interesting, just like the main result, it is really surprising that such elementary (but very tricky) methods work in this deep problem about matrices with forbidden submatrices. These results are strongly related to the theory of geometric and topological graphs. Radoslav Fulek's talk (Recent progress on Hill's conjecture) was also very interesting, he described a generalization of the methods and formula of Lov\'asz et. at. that connects crossing numbers and $k$-sets to monotone (and more general) drawings. I learned about many interesting new results, methods, ideas, from the talks and from discussions, and I hope I can use some of them in my own research.

Some of the presentations contained new ideas that both gave fresh insight to an old problem I had grown frustrated with and a new problem I had not considered before.

I really enjoyed most of the talks, some of them gave me new ideas, some gave new questions that I would like to study later in my research(f.e. Alex Scott's talk was really interesting for me, since it is close to my research area). Also, I worked during the workshop with Professor Alex Scott and Bruce Reed.