Why is the lower bound of the minimum amount of points needed so that a $4$coloring leaves at least one monochromatic triangle $62$, and not $66$? A lower bound of $66$ would seem obvious, since it is $4*(171)+2$. Would an explanation that doesn't require too much knowledge on the subject clarify this?
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The lower bound of $R(3,3,3,3)$ is not $62$, the *upper* bound is $62$. Yes, $66$ is an easy upper bound, but $62$ is way better. – Casteels Mar 22 '15 at 15:03

I did not say it was the lower bound of $R(3,3,3,3)$ but rather that there will definitely be a monochromatic triangle for numbers $\geq 62$. Could you explain why? – gogogol Mar 22 '15 at 15:05

I think the problem is with "the lower bound." Or, if Casteels is right, "the upper bound." Basically, if $62$ is an upper bound, then so is $66$, and if $66$ is a lower bound, then $62$ is a lower bound. You can talk about "the greatest lower bound" or "the least upper bound," but there isn't in general one lower bound. – Thomas Andrews Mar 22 '15 at 15:05

sorry, I mean the *greatest* upper bound for $R(3,3,3,3)$ then. – gogogol Mar 22 '15 at 15:06

You're right I wasn't thinking about my terminology correctly. What I meant is that $R(3,3,3,3)$ is at most $62$ and so at most $66$. But $62$ is better. – Casteels Mar 22 '15 at 15:06

Does anyone know why? – gogogol Mar 22 '15 at 15:07

1Here's [a paper](http://www.public.iastate.edu/~ricardo/r3333global.pdf) but it's not trivial unsurprisingly: in general, improving a known good bound by even just $1$ is often cause for celebration. – Casteels Mar 22 '15 at 15:10

So do you think I could find a more accessible explanation or probably not? Because I do not understand any of this notation/terminology. – gogogol Mar 22 '15 at 15:13

It's certainly worth at least finding an example with 61 where this is not possible. That might not be so hard. – Thomas Andrews Mar 22 '15 at 15:15