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It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $\lceil \log_2 n\rceil$. (See Fact 2.5 here.)

What about when the tournament is chosen probabilistically? What are some known results about the size of the minimum dominating set?

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The probability that a given set of size $k$ is dominating equals $(1-2^{-k})^{n-k}<e^{-(n-k)2^{-k}}$. So, the expectation of the number of dominant sets of size $k$ does not exceed $$\binom{n}ke^{-(n-k)2^{-k}}\leqslant \left(\frac{en}k\right)^ke^{-(n-k)2^{-k}}\leqslant e^{-(n-k)2^{-k}+k\log(en/k)}.$$ If $2^k<\frac{n}{C\log^2n}$ for large enough constant $C$ this tends to 0. Thus a random tournament has a dominating set of size $\log_2 n$, but does not have a dominating set of size $\log n-2\log_2 \log n-c$ for certain $c$.

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