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A Breakthrough in Graph Theory - Numberphile
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⬐ egdodHere’s a nice write-up by Gil Kalai: https://gilkalai.wordpress.com/2019/05/10/sansation-in-the-m...⬐ bhlThere's an associated Quanta article that goes with the video, along with a previous HN submission for that:⬐ nxpnsv⬐ bubblesocksThat’s great, thanks for posting!I love the way in which this lady broke down the complexity of graph theory into something that anybody can understand. I mean, I understood it.⬐ lonelappde⬐ tromp> this lady
Erica Klarriech, PhDI'd like to know the best known lower bound on graph sizes that disprove the conjecture. Could there be graphs of just a few dozen nodes that disprove it? Has anyone tried to find a counterexample by brute force search?⬐ sesuximo⬐ gorgoilerid bet yes⬐ rowanG077it's NP-hard to determine the chromatic number of a graph. So I'd wager you wouldn't get far with a brute force approach.Congratulations to Numberphile on getting to pi million subscribers.
Back in the 1990s explanations like this would be pretty common on BBC 2’s Horizon. Though not nearly as detailed, Horizon wasn’t scared of tackling complex material in a way that only appealed to niche viewers. That era of television might be over but Numberphile has very much picked up the baton. The filming style of Numberphile, especially with the delivery-to-producer rather than to camera, the natural lighting, hand held filming, and the occasional interjections from behind the camera, always bring great waves of nostalgia for the days of hard BBC science reporting.⬐ SlowRobotAheadHelp me understand... if dude proved it was false with 4^10000 exponential graph, why hasn’t a smaller tensor been tested? If he could derive a solution with a large set, shouldn’t it be easy enough to test a half size set and classic sort out where the smallest solution is?⬐ walleeee⬐ arbitrageIANA graph theorist, but I don't think it works like that. Sounds like his graph G (with ~4^100 nodes, from which the 4^10000 node exponential graph and the counter-exemplary tensor product itself derive) is a bespoke construction. I don't think it's possible in practice to just drop nodes from it and test whether the condition holds for the new tensor product. There are 4^100 ways to remove r = 1 node from an n = 4^100 node graph. Increase r and the number of combinations you'd need to test shoots through the roof. Without some way to shave off (the vast majority of) candidates, the possible solution space seems way too big for brute force search.
Someone with real expertise, please tell me if I'm wrong.⬐ lonelappdeThat's right. That's a general proof technique: to assemble a structure from a bunch of parts or layers, each of which adds to or multiplies the size of the structure, and possibly the various factors may be known only by upper bounds.
So there is room to improve slightly by brute force or more careful accounting, or a lot by finding an alternate construction.Here's an explanation so we don't have to listen to numberphile try to turn this into long-winded edutainment for youtube:
A counterexample to Hedetniemi's conjecture - featuring Erica Klarreich.
This is about graph colouring. For a short explanation of the topic at hand that's pretty good, see: https://en.wikipedia.org/wiki/Hedetniemi%27s_conjecture
The point of this video is that a counterexample to Hedetniemi's Conjecture has been found.
That is, the assertion that tensor products cannot be colored with an unexpectedly small number of colors, is wrong.
You can't say this -- that the least amount of colours needed to colour a graph which is the product of two other graphs is the least number of colours needed to individually colour the original graphs.
Apologies if i didn't get this right, but let's have a discussion about it instead of shunting people on a message-board off to youtube with no descriptions provided about what this topic actually is.⬐ JshWright⬐ krickDo you go off on this same rant on every HN post? That's the way HN works... A title that links off to some other content, and an associated discussion thread.
Personally I (not the OP) found the video interesting, and at a good level for someone without a lot of context.⬐ mokus⬐ marmadukeThey may be a bit on the ranty side but I, for one, greatly appreciate the ability to read an abstract in 15 seconds rather than however long it takes to ingest a video and separate the content from the fluff.
This is also how HN works: 99% of the time, there’s enough summary and context in the comments that I always just click through to the comments first to find out whether the article/video/whatever is worth the time.⬐ Klinky>This is also how HN works: 99% of the time, there’s enough summary and context in the comments that I always just click through to the comments first to find out whether the article/video/whatever is worth the time.
Isn't that also a folly of HN? People not reading the articles and rushing to the comments to make some proclamation or put in their two cents.⬐ munk-a⬐ JshWrightExcept you can fully understand this topic in a lot less than thirty minutes - I agree that people making half-reasoned arguments is a problem but at the same time you can comprehend the contents of this conjecture and the disproving of it via a few simple statements about graphs or relational algebra.Absolutely, but the commentor could have just done that, rather than lamenting that we should be discussing it... in a discussion thread that only exists because someone posted the video here in the first place.Can you remove the complaint about edutainment so we can up vote in earnest your summary?⬐ gjs278⬐ abacadabaoh no! they didn’t phrase it the way I like so I can’t push the button!I for one appreciate the summary. Need to make an AI youtube summarizer.⬐ JshWright⬐ dangI have no problem with the summary. I think it would have been a very strong comment if it had just skipped the first and last paragraphs.⬐ catalogiaPull the subtitles with youtube-dl, then either read it (should be much faster than watching it) or throw some typical auto-summary software at it first.
Something like: `youtube-dl --skip-download --write-sub --write-auto-sub --sub-lang=en 'https://www.youtube.com/watch?v=Tnu_Ws7Llo4'`
You can use ffmpeg to convert subtitle files between different formats. SRT is easy to clean up for reading with a a few editor macros.⬐ abacadabaword, ty.Please don't be mean about people submitting a video like that. Your comment is a mean sandwich: the filling is great information but the slices at the ends are not in keeping with the spirit of this site. That's actually more important, because it relates to the long-term survival of the community.So, a single over-exaggerated counterexample to a conjecture, which doesn't really explain anything about the nature of the class of possible counterexamples? Hardly a breakthrough. Clickbait.⬐ walleeee⬐ gwfNo one could prove or disprove the conjecture for 50 years. It was considered a very hard problem by very smart people.
I don't know about you, but I've never solved any problems that have stumped people for decades.Humble brag: Stephen Hedetniemi was my first advisor at Clemson, and I took CS101 with him back in 1985. Both faculty and students were in awe of his intellect, kindness, and modesty. What a delight to stumble upon him this morning. Thanks for posting.