# Combinatorial principles at \omega_1 and \omega_2 I

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# A Model for a Very Good Scale and a Bad Scale II

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Given a supercompact cardinal $\kappa$ and a regular cardinal

$\lambda

# Proving projective determinacy

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The principle of projective determinacy, being independent from the standard axiom system of set theory, produces a fairly complete picture of the theory of "definable" sets of reals. It is an amazing fact that projective determinacy is implied by many apparently entirely unrelated statements. One has to go through inner model theory in order to prove such implications.

# The rough classification of Banach spaces

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The geometric theory of Banach spaces underwent a tremendous

development in the decade 1990-2000 with the solution of several

outstanding conjectures by Gowers, Maurey, Odell and Schlumprecht.

Their discoveries both hinted at a previously unknown richness of the

class of separable Banach spaces and also laid the beginnings of a

classification program for separable Banach spaces due to Gowers.

However, since the initial steps done by Gowers, little progress was

made on the classification program. We shall discuss some recent

advances due to V. Ferenczi and myself on this by means of Ramsey theory

and dichotomy theorems for the structure of Banach spaces. This

simultaneously allows us to answer some related questions of Gowers

concerning the quasiorder of subspaces of a Banach space under the

relation of isomorphic embeddability.

# A Model for a Very Good Scale and a Bad Scale

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Given a supercompact cardinal $\kappa$ and a regular cardinal

$\lambda

# BMM, canonical functions, and precipitous ideals.

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We discuss how BMM affects the large cardinal

structure of V as well as the size of \theta^{L(R)}. BMM proves

that V is closed under sharps (and more), and BMM plus the

existence of a precipitous ideal on \omega_1 proves that

\delta^1_2 = \aleph_2. Part of this is joint work with my

student Ben Claverie.

# The chromatic number of subgraphs of an uncountable graph

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For an uncountable graph $X$ let $S(X)$ denote

the set of chromatic numbers of subgraphs of $X$

and $I(X)$ the analogous set for induced subgraphs.

We investigate the properties of $I(X)$ and $S(X)$.