# Minimum dominating sets in tournaments

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?

The probability that a given set of size $$k$$ is dominating equals $$(1-2^{-k})^{n-k}. So, the expectation of the number of dominant sets of size $$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$$.