In 1984 Shelah showed the consistency of $\aleph_1=\mathfrak b<\mathfrak s=\mathfrak a=\aleph_2$, however the original paper is hard to read, because of the old typesetting.
Is there a more modern exposition of this result in some paper or textbook?
Looking for a recent reference for a result of Shelah
reference-requestset-theory
Related Solutions
Well, I mean OT is a vast research area. I guess you will need to specify a little in which direction you wanna go. Besides, I personally wouldn't recommand Villani's opus magum as introductory literature. His other book 'topics in optimal transportation' or the one of Santambrogio are better for getting an overall idea of OT (personal opinion). But since you were asking for papers, I'd say the following four are amongst the 10 most influencial papers in OT of the last decades (which at the same time are not too hard to understand, I think. You may need however some background in PDE, stochastic analysis and/or Riemannian geometry (at least for the first 3), but that you will need in most of modern OT papers, anyway.):
Benamou, Brenier; A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
Jordan, Kinderlehrer, Otto; The Variational Formulation of the Fokker-Planck Equation
Otto; The geometry of dissipative evolution equations: the porous medium equation
Cuturi; Lightspeed Computation of Optimal Transport
$\omega$ and $\aleph_0$ are the same set (the natural numbers $\{0,1,2,3,\ldots\}$) by different names. $\omega$ stresses the fact that we see it as an ordinal number (the smallest one that is infinite) while $\aleph_0$ is a name that says it is the first (i.e. of index $0$) in the transfinite sequence of cardinal numbers. All cardinals are special ordinal numbers, which in turn are transitive sets well-ordered by $\in$. All this is standard modern set theory (assuming ZFC standard axioms, including well-foundedness). As to the continuum $\mathfrak{c}$ we can say that $\mathfrak{c} \ge \aleph_1$ (it's uncountable) but not much more. Also, there is a theorem that $\text{cf}(\mathfrak{c})$ (the cofinality) cannot be $\omega$, so that $\mathfrak{c}=\aleph_{\omega}$ is not possible; but other than that restriction it can be any uncountable aleph (consistently).
So it's more correct to say sets of cardinality $\aleph_0$ (not cardinality $\omega$, though one does see it occasionally). Ordinals are for "order things", cardinals for "sizes" of sets.
Best Answer
After some googling a proof of the consistency of $\mathfrak b<\mathfrak s$, which seems to be following Shelah's construction, can be found as theorem 4.5 in Uri Abraham's chapter of the handbook.