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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.
4
votes
Domination problem with sets
59k/140 sets suffice
Using your three equations, it is possible to get an improved upper bound of $59k/140$ (instead of $3k/7 = 60k/140$).
$$12a+8b+4c \leq 3k$$
$$ a+4b+2c \leq k$$
$$ a+b+4c \leq k$ …