In previous work, we used an ∞-categorical version of ultraproducts to show that, for a fixed height n, the symmetric monoidal ∞-categories En,p-local spectra are asymptotically algebraic in prime p. this paper, prove analogous result Kp(n)-local spectra, where Kp(n) is Morava K-theory at n and This requires tools suitable working with compactly generated non-compact unit. The equivalences that...

Introduction One possible way to express that the cube Q d = [0, 1] d has dimension d is to notice that it cannot be colored in d colors with arbitrarily small connected monochromatic components. A more precise statement was proved by Lebesgue: If the unit cube Q d is covered by closed sets with multiplicity at most d then one covering set must meet two opposite facets of Q d. An analogue of Gr...

Consider the following problem. In a school with three classes containing n students each, given that their genders are unknown, find minimum possible number of triples same-gender students—not all which from same class. Muaengwaeng asked this question and conjectured scenario occurs when boys, girls half-and-half. paper, we solve many generalizations problem including has more than classes, re...

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k − 1)-regular but not k-regular. We prove this conjecture by showing that the equation Pk−1 i=1 2 2i−1xi = “ −1 + Pk−1 i=1 2 2i−1 ” x0 has this property. This conjecture i...

We prove that, with high probability, any 2-edge-colouring of a random tournament on n vertices contains a monochromatic path of length Ω(n/ √ log n). This resolves a conjecture of Ben-Eliezer, Krivelevich and Sudakov and implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.