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LSpice
LSpice
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Let $S$ be a set with $|S|=|\mathbb{R}|$$\lvert S\rvert=\lvert\mathbb{R}\rvert$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $|S_x|=|\mathbb{R}|$$\lvert S_x\rvert=\lvert\mathbb{R}\lvert$ for each $x\in \mathbb{R}$. Suppose that

  • for any $s\in S$ we have $|\{x\in\mathbb{R}|s\in S_x\}|=2$$\lvert\{x\in\mathbb{R}\mathrel\vert s\in S_x\}\rvert=2$
  • for any $x\neq y\in \mathbb{R}$ we have $|S_x\cap S_y|=3$$\lvert S_x\cap S_y\rvert=3$.

ConsidersConsider functions $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection. They are indexed by $2^{\mathbb{R}}$ because each $s\in S$ can go in $2$ subsets. Is there $f$ such that $|f(S)\cap S_x|<\infty$$\lvert f(S)\cap S_x\rvert<\infty$ for each $x\in \mathbb{R}$?

Let $S$ be a set with $|S|=|\mathbb{R}|$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $|S_x|=|\mathbb{R}|$ for each $x\in \mathbb{R}$. Suppose that

  • for any $s\in S$ we have $|\{x\in\mathbb{R}|s\in S_x\}|=2$
  • for any $x\neq y\in \mathbb{R}$ we have $|S_x\cap S_y|=3$.

Considers functions $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection. They are indexed by $2^{\mathbb{R}}$ because each $s\in S$ can go in $2$ subsets. Is there $f$ such that $|f(S)\cap S_x|<\infty$ for each $x\in \mathbb{R}$?

Let $S$ be a set with $\lvert S\rvert=\lvert\mathbb{R}\rvert$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $\lvert S_x\rvert=\lvert\mathbb{R}\lvert$ for each $x\in \mathbb{R}$. Suppose that

  • for any $s\in S$ we have $\lvert\{x\in\mathbb{R}\mathrel\vert s\in S_x\}\rvert=2$
  • for any $x\neq y\in \mathbb{R}$ we have $\lvert S_x\cap S_y\rvert=3$.

Consider functions $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection. They are indexed by $2^{\mathbb{R}}$ because each $s\in S$ can go in $2$ subsets. Is there $f$ such that $\lvert f(S)\cap S_x\rvert<\infty$ for each $x\in \mathbb{R}$?

added 97 characters in body
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n_struct
n_struct
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Let $S$ be a set with $|S|=|\mathbb{R}|$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $|S_x|=|\mathbb{R}|$ for each $x\in \mathbb{R}$. Suppose that

  • for any $s\in S$ we have $|\{x\in\mathbb{R}|s\in S_x\}|=2$
  • for any $x\neq y\in \mathbb{R}$ we have $|S_x\cap S_y|=3$.

Is there a functionConsiders functions $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection and. They are indexed by $2^{\mathbb{R}}$ because each $s\in S$ can go in $2$ subsets. Is there $f$ such that $|f(S)\cap S_x|<\infty$ for alleach $x\in \mathbb{R}$?

Let $S$ be a set with $|S|=|\mathbb{R}|$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $|S_x|=|\mathbb{R}|$ for each $x\in \mathbb{R}$. Suppose that

  • for any $s\in S$ we have $|\{x\in\mathbb{R}|s\in S_x\}|=2$
  • for any $x\neq y\in \mathbb{R}$ we have $|S_x\cap S_y|=3$.

Is there a function $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection and such that $|f(S)\cap S_x|<\infty$ for all $x\in \mathbb{R}$?

Let $S$ be a set with $|S|=|\mathbb{R}|$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $|S_x|=|\mathbb{R}|$ for each $x\in \mathbb{R}$. Suppose that

  • for any $s\in S$ we have $|\{x\in\mathbb{R}|s\in S_x\}|=2$
  • for any $x\neq y\in \mathbb{R}$ we have $|S_x\cap S_y|=3$.

Considers functions $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection. They are indexed by $2^{\mathbb{R}}$ because each $s\in S$ can go in $2$ subsets. Is there $f$ such that $|f(S)\cap S_x|<\infty$ for each $x\in \mathbb{R}$?

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n_struct
n_struct
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Include each point of continuum in a subset so that each subset gets finitely many points

Let $S$ be a set with $|S|=|\mathbb{R}|$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $|S_x|=|\mathbb{R}|$ for each $x\in \mathbb{R}$. Suppose that

  • for any $s\in S$ we have $|\{x\in\mathbb{R}|s\in S_x\}|=2$
  • for any $x\neq y\in \mathbb{R}$ we have $|S_x\cap S_y|=3$.

Is there a function $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection and such that $|f(S)\cap S_x|<\infty$ for all $x\in \mathbb{R}$?