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Mad families of Gowers' infinite block sequences

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abstract

Call a subset of $\mathbf{FIN}_k$ small if it does not contain a copy of $\langle{A\rangle}$ for some infinite block sequence $A \in \mathbf{FIN}_k^{[\infty]}$. Gowers' $\mathbf{FIN}_k$ theorem asserts that the set of small subsets of $\mathbf{FIN}_k$ forms an ideal, so it is sensible to consider almost disjoint families of $\mathbf{FIN}_k$ with respect to the ideal of small subsets of $\mathbf{FIN}_k$. We shall show that $\mathfrak{a}_{\mathbf{FIN}_k}$, the smallest possible cardinality of an infinite mad family of $\mathbf{FIN}_k$, is uncountable.

fields

cs.CL 1

years

2025 1

verdicts

REJECT 1

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