
A Breakthrough in Graph Theory [video] - kgwxd
https://www.youtube.com/watch?v=Tnu_Ws7Llo4
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gwf
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.

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egdod
Here’s a nice write-up by Gil Kalai:
[https://gilkalai.wordpress.com/2019/05/10/sansation-in-
the-m...](https://gilkalai.wordpress.com/2019/05/10/sansation-in-the-morning-
news-yaroslav-shitov-counterexamples-to-hedetniemis-conjecture/)

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bhl
There's an associated Quanta article that goes with the video, along with a
previous HN submission for that:

[1] [https://www.quantamagazine.org/mathematician-disproves-
hedet...](https://www.quantamagazine.org/mathematician-disproves-hedetniemis-
graph-theory-conjecture-20190617/)

[2]
[https://news.ycombinator.com/item?id=20203707](https://news.ycombinator.com/item?id=20203707)

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bubblesocks
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
> this lady

Erica Klarriech, PhD

[https://en.m.wikipedia.org/wiki/Erica_Klarreich](https://en.m.wikipedia.org/wiki/Erica_Klarreich)

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tromp
I'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?

~~~
rowanG077
it'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.

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gorgoiler
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.

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SlowRobotAhead
Help 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
IANA 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.

~~~
lonelappde
That'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.

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krick
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
No 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.

