Author:
Girão António,Illingworth Freddie,Michel Lukas,Savery Michael,Scott Alex
Abstract
AbstractColour the edges of the complete graph with vertex set $${\{1, 2, \dotsc , n\}}$$
{
1
,
2
,
⋯
,
n
}
with an arbitrary number of colours. What is the smallest integer f(l, k) such that if $$n > f(l,k)$$
n
>
f
(
l
,
k
)
then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that $$f(l, k) = l^{k - 1}$$
f
(
l
,
k
)
=
l
k
-
1
and proved this for $$l \geqslant (3 k)^{2 k}$$
l
⩾
(
3
k
)
2
k
. We prove the conjecture for $$l \geqslant k^3 (\log k)^{1 + o(1)}$$
l
⩾
k
3
(
log
k
)
1
+
o
(
1
)
and establish the general upper bound $$f(l, k) \leqslant k (\log k)^{1 + o(1)} \cdot l^{k - 1}$$
f
(
l
,
k
)
⩽
k
(
log
k
)
1
+
o
(
1
)
·
l
k
-
1
. This reduces the gap between the best lower and upper bounds from exponential to polynomial in k. We also generalise some of these results to the tournament setting.
Publisher
Springer Science and Business Media LLC
Reference8 articles.
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2. Bucić, M., Letzter, S., Sudakov, B.: Monochromatic paths in random tournaments. Random Struct. Algorithms 54(1), 69–81 (2019). https://doi.org/10.1002/rsa.20780
3. de Bruijn, N.G., van Ebbenhorst Tengbergen, Ca., Kruyswijk, D.: On the set of divisors of a number. Nieuw Archief voor Wiskunde 2(23), 191–193 (1951)
4. Erdős, P., Rado, R.: A combinatorial theorem. J. Lond. Math. Soc. s1-25(4), 249–255 (1950). https://doi.org/10.1112/jlms/s1-25.4.249
5. Jiang, T., Mubayi, D.: New upper bounds for a canonical Ramsey problem. Combinatorica 20(1), 141–146 (2000). https://doi.org/10.1007/s004930070037
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