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Fri, 15 Dec 2017
Wasteful and frugal proofs in Ramsey theory
This math.se question asks how to show that, among any 11 integers, one can find a subset of exactly six that add up to a multiple of 6. Let's call this “Ebrahimi’s theorem”. This was the last thing I read before I put away my phone and closed my eyes for the night, and it was a race to see if I would find an answer before I fell asleep. Sleep won the race this time. But the answer is not too hard.
Here is a randomly-generated example: $$3\quad 17\quad 35\quad 42\quad 44\quad 58\quad 60\quad 69\quad 92\quad 97\quad 97$$ Looking at the first 5 numbers !!3\ 17\ 35\ 42\ 44!! we see that on division by 3 these have remainders !!0\ 2\ 2\ 0\ 2!!. The remainder !!2!! is there three times, so we choose those three numbers !!\langle17\ 35\ 44\rangle!!, whose sum is a multiple of 3, and set them aside. Now we take the leftover !!3!! and !!42!! and supplement them with three more unused numbers !!58\ 60\ 69!!. The remainders are !!0\ 0\ 1\ 0\ 0!! so we take !!\langle3\ 42\ 60\rangle!! and set them aside as a second group. Then we take the five remaining unused numbers !!58\ 69\ 92\ 97\ 97!!. The remainders are !!1\ 0\ 2\ 1\ 1!!. The first three !!\langle 58\ 69\ 92\rangle!!have all different remainders, so let's use those as our third group. The three groups are now !! \langle17\ 35\ 44\rangle, \langle3\ 42\ 60\rangle, \langle58\ 69\ 92\rangle!!. The first one has an even sum and the second has an odd sum. The third group has an odd sum, which matches the second group, so we choose the second and third groups, and that is our answer: $$3\qquad 42\qquad 60\qquad 58 \qquad 69 \qquad 92$$ The sum of these is !!324 = 6\cdot 54!!. This proves that 11 input numbers are sufficient to produce one output set of 6 whose sum is a multiple of 6. Let's write !!E(n, k)!! to mean that !!n!! inputs are enough to produce !!k!! outputs. That is, !!E(n, k)!! means “any set of !!n!! numbers contains !!k!! distinct 6-element subsets whose sum is a multiple of 6.” Ebrahimi’s theorem, which we have just proved, states that !!E(11, 1)!! is true, and obviously it also proves !!E(n, 1)!! for all larger !!n!!. I would like to consider the following questions:
I am specifically not asking whether !!E(10, 1)!! or !!E(11, 2)!! are actually false. There are easy counterexamples that can be found without reference to the proof above. What I want to know is if the proof, as given, contains nontrivial information about these questions. The reason I think this is interesting is that I think, upon more careful examination, that I will find that the proof above does prove at least one of these, perhaps with a very small bit of additional reasoning. But there are many similar proofs that do not work this way. Here is a famous example. Let !!W(n, k)!! be shorthand for the following claim:
Then:
!!W()!!, like !!E()!!, is monotonic: van der Waerden's theorem trivially implies !!W(n, 1)!! for all !!n!! larger than 325. Does it also imply that !!W(n, 1)!! is false for smaller !!n!!? No, not at all; this is actually untrue. Does it also imply that !!W(325, k)!! is false for !!k>1!!? No, this is false also. Van der Waerden's theorem takes 325 inputs (the integers) and among them finds one output (the desired set of three). But this is extravagantly wasteful. A better argument shows that only 9 inputs were required for the same output, and once we know this it is trivial that 325 inputs will always produce at least 36 outputs, and probably a great many more. Proofs of theorems in Ramsey theory are noted for being extravagant in exactly this way. But the proof of Ebrahimi's theorem is different. It is not only frugal, it is optimally so. It uses no more inputs than are absolutely necessary. What is different about these cases? What is the source the frugality of the proof of Ebrahimi’s theorem? Is there a way that we can see from examination of the proof that it will be optimally frugal? Ebrahimi’s theorem shows !!E(11, 1)!!. Suppose instead we want to show !!E(n, 2)!! for some !!n!!. From Ebrahimi’s theorem itself we immediately get !!E(22, 2)!! and indeed !!E(17, 2)!!. Is this the best we can do? (That is, is !!E(16, 2)!! false?) I bet it isn't. If it isn't, what went wrong? Or rather, what went right in the !!k=1!! case that stopped working when !!k>1!!? I don't know. [Other articles in category /math] permanent link Sat, 09 Dec 2017The Volokh Conspiracy is a frequently-updated blog about legal issues. It reports on interesting upcoming court cases and recent court decisions and sometimes carries thoughtful and complex essays on legal theory. It is hosted by, but not otherwise affiliated with, the Washington Post. Volokh periodically carries a “roundup of recent federal court decisions”, each with an intriguing one-paragraph summary and a link to the relevant documents, usually to the opinion itself. I love reading federal circuit court opinions. They are almost always carefully thought out and clearly-written. Even when I disagree with the decision, I almost always concede that the judges have a point. It often happens that I read the decision and say “of course that is how it must be decided, nobody could disagree with that”, and then I read the dissenting opinion and I say exactly the same thing. Then I rub my forehead and feel relieved that I'm not a federal circuit court judge. This is true of U.S. Supreme Court decisions also. Back when I had more free time I would sometimes visit the listing of all recent decisions and pick out some at random to read. They were almost always really interesting. When you read the newspaper about these decisions, the newspaper always wants to make the issue simple and usually tribal. (“Our readers are on the (Red / Blue) Team, and the (Red / Blue) Team loves mangel-wurzels. Justice Furter voted against mangel-wurzels, that is because he is a very bad man who hates liberty! Rah rah team!”) The actual Supreme Court is almost always better than this. For example we have Clarence Thomas's wonderful dissent in the case of Gonzales v. Raich. Raich was using marijuana for his personal medical use in California, where medical marijuana had been legal for years. The DEA confiscated and destroyed his supplier's plants. But the Constitution only gives Congress the right to regulate interstate commerce. This marijuana had been grown in California by a Californian, for use in California by a Californian, in accordance with California law, and had never crossed any state line. In a 6–3 decision, the court found that the relevant laws were nevertheless a permitted exercise of Congress's power to regulate commerce. You might have expected Justice Thomas to vote against marijuana. But he did not:
Thomas may not be a fan of marijuana, but he is even less a fan of federal overreach and abuse of the Commerce Clause. These nine people are much more complex than the newspapers would have you believe. But I am digressing. Back to Volokh's federal court roundups. I have to be careful not to look at these roundups when I have anything else that must be done, because I inevitably get nerdsniped and read several of them. If you enjoy this kind of thing, this is the kind of thing you will enjoy. I want to give some examples, but can't decide which sound most interesting, so here are three chosen at random from the most recent issue:
[ Addendum 20171213: Volokh has just left the Washington Post, and moved to Reason, citing changes in the Post's paywall policies. ] [Other articles in category /law] permanent link Fri, 08 Dec 2017I drink a lot of coffee at work. Folks there often make a pot of coffee and leave it on the counter to share, but they never make decaf and I drink a lot of decaf, so I make a lot of single cups of decaf, which is time-consuming. More and more people swear by the AeroPress, which they say makes single cups of excellent coffee very quickly. It costs about $30. I got one and tried it out. The AeroPress works like this: There is a cylinder, open at the top, closed but perforated at the bottom. You put a precut circle of filter paper into the bottom and add ground coffee on top of it. You put the cylinder onto your cup, then pour hot water into the cylinder. So far this is just a regular single-cup drip process. But after a minute, you insert a plunger into the cylinder and push it down gently but firmly. The water is forced through the grounds and the filter into the cup. In theory the press process makes better coffee than drip, because there is less opportunity to over-extract. The AeroPress coffee is good, but I did not think it tasted better than drip. Maybe someone else, fussier about coffee than I am, would be more impressed. Another the selling points is that the process fully extracts the grounds, but much more quickly than a regular pourover cone, because you don't have to wait for all the dripping. One web site boasts:
It does shorten the brew time. But you lose all the time again washing out the equipment. The pourover cone is easier to clean and dry. I would rather stand around watching the coffee drip through the cone than spend the same amount of time washing the coffee press. The same web site says:
This didn't work for me. I can't put it in my desk because it is still wet and it is difficult to dry. So it sits on a paper towel on top of my desk, taking up space and getting in the way. The cone dries faster. The picture above makes it look very complicated, but the only interesting part itself is the press itself, shown at upper left. All the other stuff is unimportant. The intriguing hexagon thing is a a funnel you can stick in the top of the cylinder if you're not sure you can aim the water properly. The scoop is a scoop. The flat thing is for stirring the coffee in the cylinder, in case you don't know how to use a spoon. I threw mine away. The thing on the right is a holder for the unused paper filters. I suspect they were afraid people wouldn't want to pay $30 for just the press, so they bundled in all this extra stuff to make it look like you are getting more than you actually are. In the computer biz we call this “shovelware”. My review: The AeroPress gets a solid “meh”. You can get a drip cone for five bucks. The advantages of the $30 AeroPress did not materialize for me, and are certainly not worth paying six times as much. [Other articles in category /tech] permanent link Thu, 07 Dec 2017As I mentioned before, I have started another
blog, called
I plan to continue to post monthly summaries here. [Other articles in category /meta/shitpost] permanent link Fri, 01 Dec 2017
Slaughter electric needle injector
[ This article appeared yesterday on At the end of the game Portal, one of the AI cores you must destroy starts reciting GLaDOS's cake recipe. Like GLaDOS herself, it starts reasonably enough, and then goes wildly off the rails. One of the more memorable ingredients from the end of the list is “slaughter electric needle injector”. I looked into this a bit and I learned that there really is a slaughter electric needle injector. It is not nearly as ominous as it sounds. The needles themselves are not electric, and it has nothing to do with slaughter. Rather, it is a handheld electric-powered needle injector tool that happens to be manufactured by the Slaughter Instrument Company, Inc, founded more than a hundred years ago by Mr. George Slaughter. Slaughter Co. manufactures tools for morticians and enbalmers
preparing bodies for burial. The electric needle
injector
is one such tool; they also manufacture a cordless electric needle
injector,
mentioned later as part of the same cake recipe.
The needles themselves are quite benign. They are small, with delicate six-inch brass wires attached, and cost about twenty-five cents each. The needles and the injector are used for securing a corpse's mouth so that it doesn't yawn open during the funeral. One needle is injected into the upper jaw and one into the lower, and then the wires are twisted together, holding the mouth shut. The mortician clips off the excess wire and tucks the ends into the mouth. Only two needles are needed per mouth. There are a number of explanatory videos on YouTube, but I was not able to find any actual demonstrations. [Other articles in category /tech] permanent link |