Archive for August, 2017

Intransitive dice VII — aiming for further results

August 12, 2017

While Polymath13 has (barring a mistake that we have not noticed) led to an interesting and clearly publishable result, there are some obvious follow-up questions that we would be wrong not to try to answer before finishing the project, especially as some of them seem to be either essentially solved or promisingly close to a solution. The ones I myself have focused on are the following.

1. Is it true that if two random elements $A$ and $B$ of $[n]^n$ are chosen, then $A$ beats $B$ with very high probability if it has a sum that is significantly larger? (Here “significantly larger” should mean larger by $f(n)$ for some function $f(n)=o(n^{3/2})$ — note that the standard deviation of the sum has order $n^{3/2}$, so the idea is that this condition should be satisfied one way or the other with probability $1-o(1)$).
2. Is it true that the stronger conjecture, which is equivalent (given what we now know) to the statement that for almost all pairs $(A,B)$ of random dice, the event that $A$ beats a random die $C$ has almost no correlation with the event that $B$ beats $C$, is false?
3. Can the proof of the result obtained so far be modified to show a similar result for the multisets model?

The status of these three questions, as I see it, is that the first is basically solved — I shall try to justify this claim later in the post, for the second there is a promising approach that will I think lead to a solution — again I shall try to back up this assertion, and while the third feels as though it shouldn’t be impossibly difficult, we have so far made very little progress on it, apart from experimental evidence that suggests that all the results should be similar to those for the balanced sequences model. [Added after finishing the post: I may possibly have made significant progress on the third question as a result of writing this post, but I haven’t checked carefully.]
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