Anti-Adenovirus Polyclonal antibody

We study Ramsey's theorem for pairs and two colours in the context of the theory of alpha-large sets introduced by Ketonen and Solovay. We prove that any 2-colouring of pairs from an omega(300n)-large set admits an omega(n)-large homogeneous set. We explain how a formalized version of this bound gives a more direct proof, and a strengthening, of the recent result of Patey and Yokoyama (Adv Math 330: 1034-1070, 2018) stating that Ramsey's theorem for pairs and two colours is for all Sigma(0)(2)-conservative over the axiomatic theory RCA(0) (recursive comprehension).

We improve the Gagliardo-Nirenberg inequality phi. Lq (Rn) = C .. Lr (Rn)L-( 1 q - n-2 2n) (.. Lr (Rn)), r = 2, 0 < q < 2n (n-2)+, L generalizing L(s) = ln-1 2s for 0 < s < 1, from Fila and Winkler (Adv Math 357, 2019. https:// doi.org/10.1016/j.aim.2019.106823) for rapidly decaying functions (.. W1,2(Rn)\{0} with finite K = Rn L(|.|)) by specifying the dependence of C on K and by allowing arbitrary r >= 1.