Nóra Frankl: On problems related to the chromatic number of the space with the maximum and Euclidean norm

In the first half of the talk we consider some results about the maximum density of a subset of Z that does not contain a congruent copy of a given finite set H. In particular, we prove a conjecture of Schmidt and Tuller about the |H|=3 case. These one dimensional results are related to the chromatic number of d-dimensional space with the max norm. This part is joint work with Andrey Kupavskii and Arsenii Sagdeev. In the second half of the talk we consider questions that are related to the chromatic number of the Euclidean plane. Let S be a finite subset of R^d and p be a point of S. In a colouring of R^d we call the pair (S,p) almost monochromatic, if S\{p} is monochromatic, but S is not. We consider questions about finding almost monochromatic similar copies of a given pair (S,p) under certain restrictions on the colouring. These questions were originally motivated by finding a human verifiable proof of chi(R^2)>4, but are also interesting on their own right. This part is joint work with Tamás Hubai and Dömötör Pálvölgyi.