Improved Lower Bounds for Van Der Waerden Numbers
Recently, Ben Green proved that the two-color van der Waerden number ω (3, k ) is bounded from below by k b 0 ( k ) where b 0 ( k ) = c 0 ( log k log log k ) 1 / 3 . We prove a new lower bound of k b ( k ) with b ( k ) = c log k log log k . This is done by modifying Green’s argument, replacing a com...
Gespeichert in:
| Veröffentlicht in: | Combinatorica (Budapest. 1981) 2022-12, Vol.42 (Suppl 2), p.1231-1252 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Zusammenfassung: | Recently, Ben Green proved that the two-color van der Waerden number
ω
(3,
k
) is bounded from below by
k
b
0
(
k
)
where
b
0
(
k
)
=
c
0
(
log
k
log
log
k
)
1
/
3
. We prove a new lower bound of
k
b
(
k
)
with
b
(
k
)
=
c
log
k
log
log
k
. This is done by modifying Green’s argument, replacing a complicated result about random quadratic forms with an elementary probabilistic result. |
|---|---|
| ISSN: | 0209-9683 1439-6912 |
| DOI: | 10.1007/s00493-022-4925-2 |