Does there exist a subset \(A\) of \(\mathbb{R}^2\) such that \( \lvert A\cap L\rvert=2\) for every straight line \(L\)?

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Does there exist a subset \(A\) of \(\mathbb{R}^2\) such that \( \lvert A\cap L\rvert=2\) for every straight line \(L\)?

Prove or disprove that every uncountable collection of subsets of a countably infinite set must have two members whose intersection has at least 2014 elements.

Let A_{1}, A_{2}, A_{3}, …, A_{n} be finite sets such that |A_{i}| is odd for all 1≤i≤n and |A_{i}∩A_{j}| is even for all 1≤i<j≤n. Prove that it is possible to pick one element a_{i} in each set A_{i} so that a_{1}, a_{2}, …,a_{n} are distinct.

Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {a_{n}} satisfying \(\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1\). Prove that there is a one-to-one function from the set of all real numbers to S.