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A couple of years ago, I was astonished and delighted to learn about Laver tables, a sequence (indexed on $n$) of Cayley-like tables for a binary operation $\star$ on numbers $i,j\leq 2^n$ that satisfies $p\star 1\stackrel{\text{def}}{=}p+1\bmod 2^n$ and $p\star (q\star r)\stackrel{\text{def}}{=}(p\star q)\star(p\star r)$. As the Wikipedia page notes, these have connections with elementary embeddings of cardinals (and apparently some connections with representations of braid groups as well, though I know less about that).

In particular, it's known that the top 'row' of the table - the list of entries $1\star q$ - is periodic for each $n$, with period $2^k$ for some $k\lt n$. It's relatively straightforward to show that this period sequence is nondecreasing (larger tables project onto smaller ones). All the tables that have been calculated have period 16 or less, and it's known that the smallest $n$ (if any) with a period larger than 16 is titanic. On the other hand, the Wikipedia page notes that the sequence of periods is 'known' to be unbounded - but only under the assumption of one of the strongest large-cardinal hypotheses known!

It's this last result that I'm hoping for an update on; is anything 'new' known about the unboundedness of the period sequence? Has it been shown to hold unconditionally? If not, is there any revised upper or lower bound on the hypothesis needed for unboundedness? I've seen Dehornoy's result that the unboundedness can't be proven in PRA, but has it been proven independent of PA itself (or even ZFC, e.g. needing some large-cardinal hypothesis) yet?

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I'm wondering whether it's provable in ZFC that there is a row with period 32 even. The tables are astonishing indeed. – user43208 Apr 11 '15 at 16:54

The problem about the unboundedness is still completely open. Furthermore, no lower bound for the rate of growth of the function $n\mapsto o_{n}(1)$ has been calculated since the 90's when Dougherty has shown that this function grows only slightly faster than the Ackermann function. Dougherty has stated that “pushing the lower bound on the growth rate of the number $F(n)$ of critical points below $\kappa_{n}$, to a function beyond $F_{\omega+1}$, will probably require a new idea” in 1 where he has proven that $F(n)$ grows faster than the Ackermann function. This seems to be a very difficult problem regardless of whether this problem is independent of PA or not.

There does not seem to be any original research publication that focuses on Laver tables published from 2000 to 2013 (but that is about to change ).

1 [Critical points in an algebra of elementary embeddings]1

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