You Will Never Escape These Sequences

[@WrathofMath]

Very busy so I had to do this one over dinner, think of it like a math dinner date! Join Wrath of Math to get exclusive videos, lecture notes, and more: kzbin.info/door/yEKvaxi8mt9FMc62MHcliwjoin More math chats: kzbin.info/aero/PLztBpqftvzxXQDmPmSOwXSU9vOHgty1RO

[@poorman-trending]

That’s got to be the nastiest looking piece of pizza I’ve seen in awhile.

[@billiboi122]

My gluttonous ass would still eat it

[@SamuelBrown-g7d]

​@@billiboi122same

[@davidroddini1512]

I don’t know. It doesn’t look nasty to me 😜

[@headpenguin8758]

looks like raw meat

[@JacksonBockus]

To me the pigeonhole principle has the best combination of “incredibly obvious fact” and “incredibly unintuitive implications”. So many weird facts can be proven by creatively transforming them into applications of the pigeonhole principle.

[@ethos8863]

The 317 joke is severely underrated. I've been giggling a while after that one

[@kyay10]

Any hints on what it is? Obviously it'd reach 325, but i don't see the joke there

[@ethos8863]

@@kyay10 the joke is that the way he was going made it seem likhe was about to prove the lower bound was 325 and then he casually says the lower bound is actually just nine, one number higher. it's funny as in "if the number is 9, why did you say 325?"

[@WrathofMath]

The joke is 1) that one would be dumb enough to suggest such a large sequence when we know a much smaller one is sufficient and 2) it's amusing foreshadowing for the actual upper bound of 325 that we prove later in the video

[@friendfrequent3330]

call this a mario bros speedrun cause why he eating a pizza?

[@WrathofMath]

very hungry, I ate an entire Shaq-a-roni pizza

[@acmhfmggru]

Yes, absolutely you should make video(s) on Ramsey's theory. There is intetest!

[@WrathofMath]

Good to know! I have to make some lecture videos on the subject still for my graph theory playlist, but they are a great candidate for these more casual video as well.

[@CY3O923]

0:00 Dam, that pizza looks good and 1:18 Can i have some?

[@WrathofMath]

all gone :(

[@omegahaxors9-11]

This is like a mathematical version of "you need at least 3 colors to make a map of countries without overlap"

[@landsgevaer]

You need 4, on a plane. Which in itself already is mathematical problem.

[@ethos8863]

I think the van der waerden problem has a very simple reason it must be true. It's hard to verbalize succinctly but essentially, of any given length, the number of unique ordered colorings is finite. After that length, you must either pick a new coloring or reuse an old one. If you reuse an old one, you now have the restriction that going forward, the distance between this coloring and its earlier use being d, in d turns, you cannot reuse this coloring, so you have to use a different coloring(which also must be a reuse if an old coloring). What happens is you essentially create a bunch of "mines" which limit the colorings you can pick over some interval. The outcome is that you run out of colorings and must pick a coloring which hits landmine, so to speak

[@WrathofMath]

Well said!

[@landsgevaer]

Now turn that into a proof... 😉 Like, if every next layer of mines add a probability 3^(-n) of being hit, the total probability of hitting a mine still only amounts to 50%. Or, more to the point perhaps, prove that it is impossible to smartly lay the mines such that a narrow path always remains. There is a reason why proofs are required to be rigorous. And why theorems that have already been proven often seem obviously true in hindsight. Otherwise you would be able to prove the twin prime conjecture and what not with this reasoning as well...

[@der.Schtefan]

The only colour you need is Farrow&Ball. Their magnificent colours remain unmatched!

[@bigjazbo9217]

If you like this sort of thing, you must buy "Three Pearls of Number Theory" by Khinchin. The proof of Van der Waerden's Theorem is the first of the three "pearls" in this short and very inexpensive book. Khinchin was one of the great mathematicians of his time. He presents the proof clearly and methodically. (The other "pearls" are equally compelling.)

[@WrathofMath]

I think I have heard of that book, but I don't have it - I'll definitely check it out!

[@Happy_Abe]

The proof seemed to only work because you assumed the third entry in Bi and Bi+j was a red. Otherwise we wouldn’t get a sequence from the first in Bi to the third in Bi+j and so on. It will still work but i think the case where the 1st and third entry in Bi have different colors should be addressed too.

[@henryptung]

Yeah, the proof rushed a little. Could have gone through the cases in sequence, assuming WLOG that the first entry is red: 1. Second entry red: If third is red, then internal progression. If third is blue, then use proof argument (1-2-3 instead of 1-3-5). 2. Second entry blue: If third is red, use proof argument, If third is blue, use proof argument but in decreasing order (3-2-1 instead of 1-3-5). Would better illustrate the general idea of using multiple sequences targeting the same element (in the i+2j segment) and constraining it so all color choices result in a color-matching progression. Would also demonstrate why the choice of length 5 is important (you're guaranteeing the presence of an AAB or ABB color progression).

[@Happy_Abe]

@ fully agreed, that’s great!

[@WrathofMath]

Yeah for brevity I covered that one specific case, as the others are similar. Glad to have them addressed here!

[@phyphor]

"gargantuan" ... Graham's number laughs

[@hello_hi1]

Both of the cases look like where k is 3 look just like k^r

[@txikitofandango]

Swank

[@danielrhouck]

This feels similar to the monocolor rectangle problem from 2 weeks ago. Interesting applications of the Pigeonhole Principle for coloring points

[@WrathofMath]

Pigeonhole principle is constantly useful!

[@circjit]

well i'll just color from 1-9 and color 0 blue so then i don't have that sequen... well, i'll have a +1 progression and, uhh OKAY YOU GOT ME!

[@cotlim]

3:16 Thats why 7 ate 9

[@WrathofMath]

Agreed

[@frogstud]

Reminds me of the sock drawer

[@WrathofMath]

True

[@wilTfrie]

you said: 1red 2red 3blue 4blue 5red HOLD ON 5 -2 --- 3

[@charlesmarshall7045]

Are you secretly sponsored by fluorescent marker makers? :)

[@WrathofMath]

Nope, just Papa Johns!

[@ciCCapROSTi]

I love tinkering like this, does it actually have any real world use?

[@empathogen75]

It gets used in computer science in a variety of ways, as are most of the interesting results in number theory, eventually.

[@Gestersmek]

I'm assuming the answer is yes, but is it a coincidence that the Van Der Waerden numbers of W(2,3) and W(3,3) happen to be equal to 3² and 3³ respectively?

[@TyroRNG]

Given that W(4,3)=76 and not 81 I think it can be assumed to be a coincidence.

[@Dooge]

This is the law of small numbers, there's just so few integers at the start of the number line that patterns will arise where there aren't any / there might be a more complex predictor that we haven't found

[@SonniXD]

My thoughts while watching 💭 Super Mario and Pizza 🍕 ... 🤔 What is this video actually about???

[@WrathofMath]

😂 Ramsey Theory!

[@vindi167]

167 likes (it is the number of the

[@matthijshebly]

πz²a

[@vampire_catgirl]

This is cool, but may I take a bite of your pizza?

[@WrathofMath]

Unfortunately I ate the whole thing :(

[@hazevthewolf178]

I like my pepperoni pizza with a topping a pineapple on it.

[@tomkerruish2982]

HERESY!

[@hazevthewolf178]

@@tomkerruish2982 :}

[@hazevthewolf178]

​@@tomkerruish2982But tastsy.

[@hazevthewolf178]

​@@tomkerruish2982:)

[@WrathofMath]

Well you know i have a rich history with pineapple, but I've actually never had a pizza with pineapple. On the other hand I have had pizza with apple on it, and it was one of my favorite pizzas. Short Rib BBQ, with short rib, bbq sauce, onions I think, and candied apples.

[@hancocki]

mmmmmmm pizza

[@tristanridley1601]

"translate to English" lol

[@This_used_to_be_my_moms]

5 hours ago

[@AavyanTiwari]

Something 'bout that pizza Fifth!

[@WrathofMath]

Shaq-a-roni, looking for that papa johns sponsorship