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research
Monochromatic loose paths in multicolored
k
k
k
-uniform cliques
Authors
Andrzej Dudek
Andrzej RuciΕski
Publication date
26 September 2019
Publisher
Doi
View
on
arXiv
Abstract
For integers
k
β₯
2
k\ge 2
k
β₯
2
and
β
β₯
0
\ell\ge 0
β
β₯
0
, a
k
k
k
-uniform hypergraph is called a loose path of length
β
\ell
β
, and denoted by
P
β
(
k
)
P_\ell^{(k)}
P
β
(
k
)
β
, if it consists of
β
\ell
β
edges
e
1
,
β¦
,
e
β
e_1,\dots,e_\ell
e
1
β
,
β¦
,
e
β
β
such that
β£
e
i
β©
e
j
β£
=
1
|e_i\cap e_j|=1
β£
e
i
β
β©
e
j
β
β£
=
1
if
β£
i
β
j
β£
=
1
|i-j|=1
β£
i
β
j
β£
=
1
and
e
i
β©
e
j
=
β
e_i\cap e_j=\emptyset
e
i
β
β©
e
j
β
=
β
if
β£
i
β
j
β£
β₯
2
|i-j|\ge2
β£
i
β
j
β£
β₯
2
. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let
R
(
P
β
(
k
)
;
r
)
R(P_\ell^{(k)};r)
R
(
P
β
(
k
)
β
;
r
)
be the minimum integer
n
n
n
such that every
r
r
r
-edge-coloring of the complete
k
k
k
-uniform hypergraph
K
n
(
k
)
K_n^{(k)}
K
n
(
k
)
β
yields a monochromatic copy of
P
β
(
k
)
P_\ell^{(k)}
P
β
(
k
)
β
. In this paper we are mostly interested in constructive upper bounds on
R
(
P
β
(
k
)
;
r
)
R(P_\ell^{(k)};r)
R
(
P
β
(
k
)
β
;
r
)
, meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of
P
β
(
k
)
P_\ell^{(k)}
P
β
(
k
)
β
in every coloring. In particular, we show that there is a constant
c
>
0
c>0
c
>
0
such that for all
k
β₯
2
k\ge 2
k
β₯
2
,
β
β₯
3
\ell\ge3
β
β₯
3
,
2
β€
r
β€
k
β
1
2\le r\le k-1
2
β€
r
β€
k
β
1
, and
n
β₯
k
(
β
+
1
)
r
(
1
+
ln
β‘
(
r
)
)
n\ge k(\ell+1)r(1+\ln(r))
n
β₯
k
(
β
+
1
)
r
(
1
+
ln
(
r
))
, there is an algorithm such that for every
r
r
r
-edge-coloring of the edges of
K
n
(
k
)
K_n^{(k)}
K
n
(
k
)
β
, it finds a monochromatic copy of
P
β
(
k
)
P_\ell^{(k)}
P
β
(
k
)
β
in time at most
c
n
k
cn^k
c
n
k
. We also prove a non-constructive upper bound
R
(
P
β
(
k
)
;
r
)
β€
(
k
β
1
)
β
r
R(P_\ell^{(k)};r)\le(k-1)\ell r
R
(
P
β
(
k
)
β
;
r
)
β€
(
k
β
1
)
β
r
Similar works
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Last time updated on 02/12/2023