Timeline for How many ways to win a game between two teams with arbitrary player skills
Current License: CC BY-SA 4.0
5 events
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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 |
added 6 characters in body
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Mar 24 at 17:14 | history | asked | bernardorim | CC BY-SA 4.0 |