The Mystery of the Unknown "Ramsey Numbers"

Let me explain about "Ramsey numbers", an interesting unsolved mystery with recent mathematical progress, which we can visualize through colorful shapes and analogies about parties!
Also, here are a few bonus "shorts" I released since last episode (I didn't send them to notifications or subscription feeds, so you may not have seen them)
• Shorts
• Shorts
Also make sure you're tuned in to my ‪@Domotro‬ channel for my bonus videos, livestreams, and other content!
This was filmed by Carlo Trappenberg.
Special thanks to Evan Clark and to all of my Patreon supporters:
Max, George Carozzi, Peter Offut, Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Izeck, Beugul, OmegaRogue, Florian, William Hawkes, Michael Friemann, Claudio Fanelli, and Julian Zassenhaus.
(To join that list of people supporting this channel, and get cool bonus content, check out the Combo Class Patreon at / comboclass )
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If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com
In case people search any of these terms, some of the topics discussed in this episode are: Ramsey numbers (especially "diagonal Ramsey numbers") such as R(5,5), Ramsey's theorem / Ramsey theory, Graham's number, graph theory, complete graphs, the analogy sometimes called the "theorem on friends and strangers", the mathematician Paul Erdos, aliens, computers, and lots more!
If you're reading this, you must be interested in Combo Class. Make sure to leave a comment on this video so the algorithm shows it to more people :)
DISCLAIMER: Do not copy any uses of fire, sharp items, or other dangerous tools or activities you may see in this series. These videos are for educational (and entertainment) purposes.

[@ComboClass]

Hope you all enjoy! After you watch, check the description for other cool links (like a few new "shorts" I put on this channel without sending to notifications/subscriptions). Also check out my @Domotro channel for all my livestreams and bonus videos!

[@sonyamainprize6407]

R(9,9)=345-50,000

[@BaseSixBasics]

I predict that the next Ramsey number is 48, because base six :)

[@darreljones8645]

A slight error in the graphic that shows up around 6:30: The phrase "pigeonhole principle" is misspelled. A "pigeonhole principal" would likely be someone in charge of a mailroom. :)

[@ComboClass]

Good observation, you’re correct haha

[@ulalaFrugilega]

@@ComboClass totally on purpose for sure. 😂 I see a smug pigeon strutting around, organising all those holes into bunches of 5 which seems to the number a smart pigeon can count.

[@ComboClass]

@@ulalaFrugilega It was an accident, but as far as misspellings go, at least it was a humorous one haha

[@joedevitt132]

You are very smart... does that make you feel good?

[@ulalaFrugilega]

@@joedevitt132 🤣 don't you think it's good and helpful to point out mistakes?

[@gustavocortico1681]

This seems extremely useful and frankly kind of magical. Just imagine, you have a big set whose elements relate to each other. By merely knowing the size of the set you can immediately infer over the relationship between the elements.

[@SirCalculator]

Actually Ramsey numbers have been known to be quite useless among mathematicians so its funny that you say that :D But I agree that the whole concept is extremely fun and engaging

[@Madrawn]

@@SirCalculator I think it might be useful for circumstances in fields where such a connected group has special properties. Like if I had a polymer that formed by connecting to its neighbours either using a double bond or a triple bond, and I knew that three molecules that connect to each using only one type of bond will be broken down into some deadly toxin if you eat it. Then I know that polymer could never be made non-toxic no matter how clever I am and how much control I exercise over the reaction. (This example doesn't hold up to closer scrutiny without some further conditions like each molecule would have to always connect to at least 6 others which would have to be some form of lattice i think)

[@TheChemist94]

Dots and colors was much more intuitive explanation to me than high-fives.

[@ComboClass]

The colors are definitely easier to visualize. The high-fives were mostly to show how it can apply to real-world situations

[@Paul-fn2wb]

I'd die after a minute of such intense talking, you're awesome. It's rare I come back to a video I started watching a day before, but here I am! I heard about the numbers previously, but only now I understand their meaning. Thank you! Cute pets and studio!

[@krishnakrick7475]

@comboclass Nice presentation and explanation There is a pattern in Ramsey number table 15:03 . If you view the Ramsey number table diagonally, it represents a sort of reduced Pascal's triangle. By observation, it shows 1. The upper bound of Ramsey number is Pascal number. 2. Ramsey number is less than or equal to Pascal number.

[@drenz1523]

that is pretty interesting...

[@HeavyMetalMouse]

The real trick is to try and prove that this pattern continues to hold going forward. Many patterns in maths tend to hold for a small number of initial values, then break at a certain point at which the problem passes some boundary of complexity. Occasionally, that 'small number' can be ridiculously large, which is particularly annoying.

[@renerpho]

@@HeavyMetalMouse Which is a corollary of Guy's strong law of small numbers: "You can't tell by looking," and "there aren't enough small numbers to meet the many demands made of them".

[@drenz1523]

@@HeavyMetalMouse Looking at the Wikipedia page (in the Asymptotics part) apparently it is proven that an upper limit of a ramsey number is (r+s-2 r-1) which i think is also the formula for the pascals numbers but im not sure

[@stanleydodds9]

These are called the binomial coefficients, and it's quite easy to prove that they are an upper bound on the Ramsey numbers. Consider a graph with R(x-1,y) + R(x, y-1) vertices. Pick any vertex, and say it has r red edges and b blue edges connected to it. If r

[@aslkdjfalsdkjfasldkfj]

my new favorite channel after the eggs and complexity and this

[@PotatoSofi]

This proves that, in every party with only one person, that person will always high five themselves.

[@popularmisconception1]

or not.

[@boxmanatee]

These videos are just getting better and better.

[@stickmcskunky4345]

Tonight I am ready to learn about this cool unsolved problem in mathematics.

[@joedevitt132]

Said no one ever. This shit will send you phucking mad, as you can clearly see in most of his videos, it takes a certain type to pull it off... Mathematics, not even once.

[@lyrimetacurl0]

I wonder what the 1919th Busy Beaver number is.

[@hkayakh]

Oh so Ramsey numbers are that time travel riddle by Ted Ed

[@gregorydessingue5625]

I work in a space with no cell phones, and no internet in the computer I use. You’ve inspired me to play with the calculator occasionally on breaks 😅

[@cube2fox]

This reminds me of VC dimension, a complex concept from statistical learning theory.

[@TheCookiePup]

I found a similar sequence "A089424" on the OEIS, but it becomes invalid since 1580 is outside of the 282-1532 range.

[@TheCookiePup]

Possibly better comparisons: A241208 (divide by 2) A336127 (skip first 3 terms, divide by 8) A174470 (skip first 2 terms, divide by 9) I was searching OEIS for more instances of 1n, 2n, 6n, 18n where the next term is somewhere between 43n and 48n, and the following terms continue to fit the ranges for the lower and upper bounds.

[@stickmcskunky4345]

A000957 (Fine's sequence) also matches the first four (only) diagonal Ramsey numbers, and is related to the Catalan numbers.

[@wesleydeng71]

Thanks for the good video. I have a black cat that looks exactly the same as the first one!🐈‍⬛

[@ashleyvalleyfarms3712]

I had no clue jack harlow was so passionate about math

[@stickmandaninacan]

All i heard was "the answer is a whole number less than 50" i haven't watched further, but the answer has just gotta be 42. That super computer already figured it out

[@ComboClass]

I do love the hitchhiker’s guide to the galaxy. Unfortunately with the number I was referring to here, it has been proven larger than 42. Might be 43 though

[@ulalaFrugilega]

Why is this called Combo Class? Totally enjoy it, thanks!

[@anthropomorphicpeanut6160]

Combinatorics I guess

[@ComboClass]

There is a whole history behind why I called it Combo Class and various reasons, but one of the simpler-to-explain reasons is that I like to “combine” different topics/approaches, such as mixing pure math teaching with other things in nature/philosophy/language/etc

[@ulalaFrugilega]

@@ComboClass To me, Combo has musical associations, which does go well with mathematics, of course.

[@ComboClass]

@@ulalaFrugilega There will be musical connections in future episodes for sure :)

[@ulalaFrugilega]

@@ComboClass just saw the exponents episode and was delighted by the promise of your inside-background... but saddened by the realisation that I'm too stupid to understand why 2 does not multiply like 3 does. Stuck at 4 when I expected 8... so, though I had absolutely expected correctly, how the other numbers developed, I now doubt if I really understand. Still, I do love numbers a lot, and the way you explain things at least as much, so I enjoyed myself anyway. Thanks! By the way: do you think we must have sth. fundamentally right in presenting our numbers, if the digit sum tells us so much about a number's traits? Or would there be other ways, like the Romans', that may give different clues?

[@dananichols349]

As usual, by the end of the video I have no idea what's going on, and yet... I get the feeling that I'm more enlightened than before I watched the video.

[@azimuth4850]

Interesting.

[@snakewhitcher4189]

My left hand seeks blood.

[@blueschase11]

What was those crossings of the 4 connections and from 18gon.

[@jorian_meeuse]

There was a breakthrough in ramsey numbers very recently, apparently. I'm not sure about the details, but I believe someone solved R(4,t)

[@obiobiero6498]

Im raising tis concern late but I'd like to know how the possible colourations are calculated.Anyone who knows how?

[@good.citizen]

yep ibe skipping since the back yard camp out you know smoking or hanging out at the bowling alley, although i got the fibonacci primes i need to cram derivative intergrations mod sqare root of twelve plus an accountability of extra stellar brownian mass motion pls thank you combo class

[@devoidsloth]

I wonder if they’re all even

[@aer0a]

If you were lucky, a computer could instantly eliminate a number

[@soupisfornoobs4081]

You should not be allowed in casinos

[@keonscorner516]

R(5,5) is 45 exactly I know it

[@thebetterone7638]

Isn’t it 54 2(1+2+6+18)

[@Justyouraveragedaeodon5]

3x+1

[@andylenk959]

Perhaps use an intiger composit list with various arrangements of 0 ie(0101011010=5 and 11010110=5) then use IF statement to eliminate, or model possibilities of the angular, vertical and horizontal lattice pathway possibilities. kzbin.info/www/bejne/m6Ocm2eoaM2epMk

[@sonyamainprize6407]

R(0,0)=0

[@wjones28]

bro I just wanted to smoke and go to sleep😭now I will smoke and have my mind blown.

[@richardforster1239]

Is it known whether R(7,7) must be smaller than R(8,8)? The ranges you gave imply that R(8,8) could be smaller but my faulty human intuition expects R(7,7) to be smaller. Similar questions exist for non diagonal numbers, so R(6,6) vs R(6,7) for example. Great videos by the way!

[@ComboClass]

Yeah as you’d guess, R(7,7) can’t be larger than R(8,8), so basically R(7,7) could only be in the > 283 part of its range if R(8,8) is also above that in its own range

[@newd9848]

I really liked the part where you pet your cat and simultaneously explain the math stuff. Very cute!

[@u_cuban]

It would be interesting to hear about some of the unintended benefits of these abstract mathematical questions. For instance, I could see Ramsey Numbers being used to predict properties of metallurgical alloys based on the geometries of their bonds within repeating microstructure shapes in the alloy (much like the tiling vid I watched previously). While I do enjoy pure maths for maths' sake, I think linking it to advancements in other fields would help make the information feel more accessible and encourage cross-disciplinary consideration of the knowledge.

[@TymexComputing]

I hope that for metallurgy its enough to find that R(5,5) to be between 43-48 :) - even for the "multi-scale" metallurgy simulations

[@CatherineKimport]

This low number answer, very hard to prove problem reminds me of the search for “God’s Number,” the maximum number of moves that an optimum solution for a Rubik’s Cube scramble can have.

[@madisonjacques8507]

I really love your videos, but my favorite part of this was how happy the kitty was when you rubbed its belly

[@claytonhiggins7526]

Love the video! But how is it that we know R(a,b) always exists?

[@ComboClass]

A theorem called Ramsey’s theorem proved that there will always be a finite solution to R(a,b)

[@rose_no]

Not First

[@godhimself1125]

13:27 oh no he’s talking about r(3,4)

[@november666]

It’s crazy that mathematicians can’t figure out what 43 - 48 is. Like bro, it’s just -5

[@chrisdecke5619]

Anybody confused

[@maynardtrendle820]

I like the cleaned-up combo class.

[@TheMagicFellow]

Nice to see some of the "Combo Mail" being used in videos from the Live streams.

[@jagoandlitefoot]

i love your cats so much 🥺🥺🥺🥺🥺

[@ComboClass]

Me too! I’ll try to feature them in more episodes :)

[@tomkerruish2982]

Naïvely, this sounds like something amenable to quantum computing. I'd really like to stress that first word.

[@OzoneTheLynx]

this feels like a problem that should be efficiently solvable using quantum computers, then again I don't actually knwo quantum informatics.

[@popularmisconception1]

This sounds certainly like a combinatorial problem. And the table, when rotated diagonally, looks a lot like a pascal triangle. So I'm pretty sure intuitively that the answer to what the R(a,b) formula is has to do with combination numbers (and multinomial coefficients for more then 2 colors) and inclusion and exclusion principle. Some thinking and maybe a conjectural formula could be devised.... lemme think...

[@hkayakh]

It’s easy just guess

[@Bingcenzo]

"What were you doing at that party?" "Uh... high fiving?"

[@snakewhitcher4189]

I was thinking the exact same thing. I Was unfortunately predisposed. No math. Only teeth.

[@thatfamiiiarnight3665]

we’re up all night to the sun we’re up all night to get some we’re up all night for good fun we’re up all night to get lucky

[@stickmcskunky4345]

I might not be a computer but I can sure as hell draw a lot of dots and lines and now.. I must.

[@ulalaFrugilega]

And not just once either... reminds of the heroine in Heinrich Böll's book "Gruppenbild mit Dame" who had started drawing an eye in such detail that every cell of the retina was visible. Impossible to do in a human lifetime, but she was working on it anyway.

[@NathanBrock-ih2ee]

Why R(3,4), you couldn't have chosen any other pair of numbers. I'm not going to tell why it's a problem, but I believe you should already know.

[@darrylschultz9395]

C'mon-nobody would hi-five anyone at a party that was part of this Ramsey experiment. Get real-they'd all head straight for the beers!

[@TymexComputing]

15:40 - the definition why diagonal Ramsey number is "diagonal" - pretty obvious but i still was a suprise :)

[@peppermann]

Wonderfully and enthusiastically presented, as always!

[@bobh6728]

There is one thing he needs to study about more. That is gravity. Things fall if there is nothing holding them up!! 🤪

[@nickcarr5724]

I mean this in the best way possible: this is the most absolutely unhinged video I have ever seen.

[@victorribera5796]

This somehow feels like those cases where they find the fifth number and that gives a complete formula for all cases

[@glarynth]

Your set would make Doc Brown feel at home

[@alexhd4747]

bro stole my cat

[@theinternetis7250]

Next episode: the Reimann Zeta Function

[@ComboClass]

Not next episode, but I do plan to make an episode about that before long :)

[@monoman4083]

ramsey ramjet

[@General12th]

Hi Combo! I like to think that isn't your cat. You're just petting random cats who wander into the shot.

[@ComboClass]

Well, although they are my (3) cats, one of them was a stray who did kinda wander into my realm and started getting pets and decided to stay over time (now is officially adopted)

[@sonyamainprize6407]

R(9,9)=345-50,000

[@rickyraj7773]

Hello I don't know if you'll read this but in any case you do I just wanted to ask for any advice that you can give me for mathematics as I'm now in jr college I am finding it do be really difficult...maybe 2-3 later our class will start calculus and coordinate geo. I'm kinda scared of these two So please if there's any advice that you can give regarding How to improve in maths

[@ComboClass]

Some advice I will always recommend is to read a lot (both from books and from online encyclopedias/articles) and take notes about different ways to describe topics or connections you find between them. And work with the topics yourself, doing personal experiments to see what you can demonstrate yourself and what further questions you think of. That’s just a few pieces of advice. Good luck!

[@rickyraj7773]

@@ComboClass Thanks for the advice

[@evenaxin3628]

Would R(2,3,3) be 7?

[@sonyamainprize6407]

Yes

[@NoOffenseAnimation]

If aliens ever come and demand the 5th Ramsey number, I'll know that domotro prophecied (?) this

[@matthijshebly]

Wasn't sure about your delivery at first, but I'm warming up to it, and your content is great :)

[@nicholasweaver2374]

Aliens: People of Earth! We demand the 5th Ramsey number! Humanity: We don’t know it! Aliens: Dang, you don't know either? That's unfortunate.

[@conrmckocoa9352]

Thanks for explaining it in multiple ways with multiple visuals and examples, that's really helpful

[@JurassicJenkins]

@0:29 ChatGPT - I couldn’t resist 💪

[@user-pr6ed3ri2k]

Wait i think numberphile discussed thiw lol but i eont 4:37 0:01* really remembermrmfkrkendjrjrjrjrjrrhrhr

[@user-pr6ed3ri2k]

Minimum size for guaranteed group

[@user-pr6ed3ri2k]

Also Ted Ed for dimensional nodes thing ido

[@user-pr6ed3ri2k]

Uddk

[@user-pr6ed3ri2k]

Reminds me of grahamnum

[@user-pr6ed3ri2k]

12:34 no don't

[@pauselab5569]

almost first

[@GameJam230]

I feel like calling them dots, or vertices on a N-gon might be a red herring. After all, you can rearrange the the dots, moving their respective connections along with the ends, and you'll end up with a new arrangement of connections, which you referred to as a different "coloration". In reality though, all you've done is visually diaplaced information that you had, but you haven't actually changed it. If that is the case, then it would imply there is some inherent cause for this which is in no way related to connecting points in a geometric structure, and that any algorithm you can use to determine if two points are connected or not will always output the same results. For example, let's say we draw a red connection between any two points which are coprime with each-other, and a blue connection between any two points which are not. This will obey the rule for all *known* numbers, as in if you have 6 numbers, whether consecutive, randomly selected, or from some specific sequence, then it will be guaranteed that either a 3-gon of blue connections or (but not exclusively) a 3-gon of red connections will exist in it. And, with having the ability to change what the numbers are without affecting the validity of the statement, it could be equated to simply rearranging the vertices on a geometric shape. In both cases we have made only minor changes to the information, but some distinct force is still causing the statement to be true, which as I said, implies that it is not something that depends on the arrangement and selection of the data, but on some inherent fact about trying to connect concepts together to begin with.

[@ComboClass]

I’m not sure where you got the “n-gon” idea, I never mentioned that. Each “coloration” I described has a clear and technical definition. Like I mentioned later in the episode, they are the distinct “graphs” on n vertices, in what’s called graph theory

[@GameJam230]

@@ComboClass I was using N-gon as a way to phrase what the visual looked like, I wasn't suggesting you called them that necessarily. You drew 6 dots close to equally spaced apart for R(3,3), which is functionally equivalent to 6 vertices for a regular hexagon. But what if I took the position of two of these points, and swapped them? (To clarify, that means I move their respective connections too. So if point 1 and 5 are connected, they'll still be connected if I swap the position of point 1 and point 3, regardless of if point 3 was connected to point 5 or not). Visually speaking, it looks like we've just drawn a brand new graph with a different arrangement of connections between points, but functionally speaking, it's the same points that are connected. I only used the terms I did to attempt to convey that this may not really be a "graph theory" problem, and instead one that can be *visualized* with graph theory. Or rather, not to suggest it isn't a graph theory problem, but instead to suggest that visualizing the problem is causing a misconception with its meaning. That's more what I'm trying to get at here- that visually representing this problem is what causes the distinction I'm noting. Whether that visualization is calling it connected points, high-fiving people at a party, or numbers which are coprime. In all 3 cases, they are different ways to represent the same problem, and all obey the same rules as the others.

[@ComboClass]

Well yeah I mentioned in the video that you can move the dots wherever and it would be called the same “graph” as long as the web of connections is the same. Whether you include just the distinct graphs or include all visual ways of doing it, the rules (like that 6 dots will always have a monochromatic triangle) are true either way. And it’s not about n-gons but it’s far easier to draw the connections when you space the vertices equidistant on an invisible circle which ends up being a polygonal shape

[@ComboClass]

And it is possible to describe this problem without using any visualizations, but it would still typically be classified as graph theory. However, graph theory is very commonly given this type of visualization because otherwise it’s much harder to work with

[@hkayakh]

1 57

[@ComboClass]

One recommendation is to read a lot, both from books and from online encyclopedias/articles, and to take notes while you are reading. Then, try to do experiments or work with the concepts yourself, seeing which portions you can demonstrate yourself and which other questions emerge when you play with the topic. And throughout that, look for connections between different topics you've encountered, and take notes about those connections. That's just a few pieces of advice. Good luck.

[@paulfoss5385]

3:59 I am of two minds over the use of the term "subgroup" here instead of "subset". On the one hand "group" has a very specific meaning in mathematics, but on the other hand mathematicians clearly named "sets" and "groups" backwards. People naturally talk about "grouping" items for defining arbitrary collections, but in abstract algebra arbitrary collections of elements don't generally form groups which have to satisfy rigid properties which are set in place. People only talk about "setting" things if they have a specific arrangement in mind, like setting the table. Every single time I try to explain groups to people I have to waste a bunch of time saying what it isn't. And this video isn't employing ideas from abstract algebra, so its fine, but to someone who has studied abstract algebra it can sound a bit off and "subset" would have worked here, although it could risk making the video sound too formal and uninviting. 🤷‍♂ My pedantry aside, great video.

[@Tletna]

I don't have an answer yet but this sort of problem seems like the kind of problem where if one thinks about it differently it might become easier or harder to solve.