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Probabilistic methods prove existence results in a nonconstructive fashion, by showing the chance of randomly selecting a solution is greater than zero.
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An extremal combinatorics problem involving column summation
If you relax the power-of-two condition, you can contrive examples for the $\beta = 1/2$ and $\alpha = 1$ case that require all columns to be used. Choose a large prime $p$. Let $A$ be the $p \times …