gowers
Reputation
17,097
Next privilege 20,000 Rep.
Access 'trusted user' tools
 Mar 30 comment Family of subsets such that there are at most two sets containing two given elements It is perhaps worth adding that the above construction is generated by two standard tricks. The first is to dualize the problem by defining $T_i$ to be the set of $k$ such that $i\in S_k$ and reformulating the conditions in terms of the $T_i$. (The main one says that the maximum intersection of any two $T_i$ is 2.) The other trick is to use graphs of polynomials to get plenty of sets with small intersections. Mar 30 comment Family of subsets such that there are at most two sets containing two given elements Thanks a lot -- I've edited it now. Mar 25 comment Family of subsets such that there are at most two sets containing two given elements Have you tried taking the characteristic functions of the $S_i$, adding them up, and looking at the $\ell_2$ norm? The condition on the $S_i$ should put a strong condition on the average inner product, and then the Cauchy-Schwarz inequality should give a bound the other way. I feel this ought to work, but can't quite be certain without writing it down. Mar 23 comment Do good math jokes exist? Approximately two and a half years later I see that I didn't write what I intended to write. I did of course intend to write "compact" -- or else the joke makes no sense. In other words, Andrew Stacey's version is what I intended (except that in my version there was just one examiner). Feb 28 comment How to mentor an exceptional high school student? I've always wondered how people know they've been downvoted. Is it by repeatedly checking the number of votes so that one catches it after it decreases and before it increases again? Or is there some more efficient method that I've been too stupid to work out? Feb 7 comment Economical hard word problem I don't know whether this suits my motivation until I've messed around with it for a while. But it looks promising, so many thanks. (Sorry for not writing this earlier -- I've been away from Mathoverflow for a while.) Dec 27 comment Economical hard word problem It looks to me as though these examples aren't quite what I'm looking for, interesting though they are. I haven't properly understood the first paper you refer to, but it looks as though it takes an arbitrary semigroup and encodes it as one with just three relations. From that I deduce that the relations are probably very complicated. I think I'm more interested in the relations being short than in there being few of them, though I'd like the latter as well. Also, if there are many relations but they can be easily grasped (as occurs in, say, the braid group on n generators) I would be happy. Dec 27 comment Economical hard word problem That might be OK, but not all such sets would be convenient. For example, I think one can probably have some fun exploring patterns that can be made with Penrose tiles, but writing anything down symbolically would be a nuisance. Dec 27 comment Economical hard word problem @Dylan, I don't know if I'm looking at the same example, but if I am, then I don't like it because it encodes another problem. Maybe I should add that as a further restriction. In fact, I think I will. Oct 18 comment Is this Ramsey-type problem an open problem? @Gerhard Paseman It's not just like an chromatic number problem -- it is asking whether the chromatic number of a certain graph is infinite. (Of course, you must be aware of this or you wouldn't have posted your comment.) Unfortunately, that reformulation doesn't seem to help much, essentially because the graph is too concrete for general facts about graph colouring to be of any use. However, I did once attend a talk that discussed the problem in those terms. Aug 3 comment More on Lebesgue non-measurability Sigh -- I should have thought harder about the question before posting it. But thanks for the answer. Aug 3 comment Lebesgue non measurability in the plane See Pietro Majer's comment below. I think it deals with your objection. Jul 20 comment Method for variable substitution in multiple summation I think you need to be clearer about what the substitution should achieve. Otherwise, the question is a bit too vague. In your examples, I don't see any great simplification resulting from the substitutions, which is what confuses me. Jul 18 comment Examples of non-abelian groups arising in nature without any natural action That's the group of affine transformations of the rationals: (i,p) sends x to px+i, so (i,p)*(j,q) sends x to p(qx+j)+i=pqx + i+pj. Jul 15 comment Lower bounding the maximum size of sets in a set family with union promise So s depends on $\epsilon$ and the sets $C_1,\dots,C_k$? Jul 14 comment Current status of Waring-Goldbach problem Sorry, I take that back -- I was thinking of the result with zeros. But I think the zeros are needed for congruence reasons and it's not clear that one couldn't use some trick. E.g. if k=6 then each kth power is 1 mod 7, except that we can throw in a few $p_i=7$s to deal with that. Jul 14 comment Current status of Waring-Goldbach problem I'm pretty sure this is known for all k but I'm afraid I don't know a reference. Jul 14 comment Lower bounding the maximum size of sets in a set family with union promise With your use of the words "given" and "let" I find it hard to work out exactly how the quantification works in your question. Would it be possible to ask it in a more formal way? Jul 13 comment a Ramsey-type question Yes it was then. Jul 13 comment a Ramsey-type question By the way, I find this cluster of related questions you are asking very appealing. (My attention was first drawn to them by Noga Alon.)