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arithmetic progression
Rainbow AP(4) in an almost equinumerous coloring ★★
Author(s): Conlon
Problem Do 4-colorings of
, for
a large prime, always contain a rainbow
if each of the color classes is of size of either
or
?
![$ \mathbb{Z}_{p} $](/files/tex/ca03dcde1fc73a1d3d1916bca138cd11161ea69a.png)
![$ p $](/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png)
![$ AP(4) $](/files/tex/abd6fa4428454b30450d94292e000c9ecaa7a4fc.png)
![$ \lfloor p/4\rfloor $](/files/tex/d906d60e3b848dd93ec8be196b73063411f71d25.png)
![$ \lceil p/4\rceil $](/files/tex/5e7562752899a961fe9ccacfd0a84316504a88c2.png)
Keywords: arithmetic progression; rainbow
Long rainbow arithmetic progressions ★★
Author(s): Fox; Jungic; Mahdian; Nesetril; Radoicic
For let
denote the minimal number
such that there is a rainbow
in every equinumerous
-coloring of
for every
Conjecture For all
,
.
![$ k\geq 3 $](/files/tex/c403b537b372cb5d16b6129dce0f0c416c1f1b31.png)
![$ T_k=\Theta (k^2) $](/files/tex/861c6fb3fc588a2247540bcd7bf32e42032ad265.png)
Keywords: arithmetic progression; rainbow
Concavity of van der Waerden numbers ★★
Author(s): Landman
For and
positive integers, the (mixed) van der Waerden number
is the least positive integer
such that every (red-blue)-coloring of
admits either a
-term red arithmetic progression or an
-term blue arithmetic progression.
Conjecture For all
and
with
,
.
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
![$ k \geq \ell $](/files/tex/739b55f8a55e492f6a853fe336e2a04b05bd1182.png)
![$ w(k,\ell) \geq w(k+1,\ell-1) $](/files/tex/783e69162d75cf0eb2d5bd5a8d8b3410b04e5107.png)
Keywords: arithmetic progression; van der Waerden
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