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4 votes
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Cardinality of splitting families

For any set $X$ let $\operatorname{SF}(X)$ be the the set of all families $\mathcal S\subseteq\mathcal P(X)$ such that, for each pair $\{x,y\}\in[X]^2$, there is a set $A\in\mathcal S$ with $|A\cap\{x,...
bof's user avatar
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8 votes

Does every linear cover contain a minimal cover?

Partial answer. I show that a minimal subcover exists in two special cases, namely, if $\mathcal C$ consists of $2$-element sets, or if $X$ is countable. Theorem 1. A cover consisting of $2$-element ...
bof's user avatar
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7 votes
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(Weakly) minimal subcovers of linear covers

The answer is yes. I will prove the contrapositive. Suppose $\mathcal C$ is a cover of a nonempty set $X$ with no weakly minimal subcover; I claim that $\mathcal C$ is nonlinear. (In fact there is an ...
bof's user avatar
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9 votes

Balancing act for infinite walks

The limit of the following sequences probably works: $A_0 = \varepsilon$ $A_i = (A_{i-1}0)^{2^{(i^2)}}A_{i-1}(1A_{i-1})^{2^{(i^2)}}$ Intuition: For every $S$, there exists some $A_i$ which is not a ...
1001's user avatar
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