Graphons are analytic objects representing limits of convergent sequences of graphs. Lovasz and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many graph densities, has a simple structure. In particu
lar, one of their conjectures would imply that every finitely forcible graphon has a weak $varepsilon$-regular partition with the number of parts bounded by a polynomial in $varepsilon^{-1}$. We construct a finitely forcible graphon $W$ such that the number of parts in any weak $varepsilon$-regular partition of $W$ is at least exponential in $varepsilon^{-2}/2^{5log^*varepsilon^{-2}}$. This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons.
We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertic
es. Improving an upper bound of Axenovich and Iverson, we show that maxR(n,K_m,K_4) <= n^{3/2}sqrt{2m} for all m >= 3. Further, we discuss a possible way to improve their lower bound on maxR(n,K_4,K_4) based on incidence graphs of finite projective planes.
