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Mar 25 at 10:05 comment added bernardorim Thanks very much for the comments, @IlyaBogdanov! If question 2 is restricted to the case of odd $n$, is the answer positive? (low odd integers $n$ can be checked by computer, but a general argument is certainly better).
Mar 25 at 7:36 comment added Ilya Bogdanov As for Question 2, the answer is a clear no when $k=n/2$. Either all skills are equal, and then $w_k={n\choose k}$, or not, and then there are many losing teams of $k$ members, so ${n\choose k}-1$ is not achieved, if $k>1$.
Mar 25 at 7:33 comment added Ilya Bogdanov As for Question 1, any two winning teams $B$ with $n-k$ members should intersect, so by Erdos--Ko--Rado there are at most ${n-1\choose n-k-1}={n-1\choose k}$ of them. Therefore, the number of non-losing teams $A$ is at least ${n\choose k}-{n-1\choose k}={n-1\choose k-1}$.
Mar 24 at 18:01 history edited bernardorim CC BY-SA 4.0
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Mar 24 at 17:14 history asked bernardorim CC BY-SA 4.0