On the maximum number of integer colourings with forbidden monochromatic sums
Let f ( n , r ) denote the maximum number of colourings of A ⊆ { 1 , … , n } with r colours such that each colour class is sum-free. Here, a sum is a subset { x , y , z } such that x + y = z . We show that f ( n , 2 ) = 2 ⌈ n / 2 ⌉ , and describe the extremal subsets. Further, using linear...
| Main Authors: | , , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Electronic Journal of Combinatorics
2021
|
| _version_ | 1797069610533519360 |
|---|---|
| author | Liu, H Sharifzadeh, M Staden, K |
| author_facet | Liu, H Sharifzadeh, M Staden, K |
| author_sort | Liu, H |
| collection | OXFORD |
| description | Let
f
(
n
,
r
)
denote the maximum number of colourings of
A
⊆
{
1
,
…
,
n
}
with
r
colours such that each colour class is sum-free. Here, a sum is a subset
{
x
,
y
,
z
}
such that
x
+
y
=
z
. We show that
f
(
n
,
2
)
=
2
⌈
n
/
2
⌉
, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of
f
(
n
,
r
)
for
r
⩽
5
. Similar results were obtained by Hán and Jiménez in the setting of finite abelian groups. |
| first_indexed | 2024-03-06T22:27:01Z |
| format | Journal article |
| id | oxford-uuid:56ff0042-85a1-488e-a83b-5571a00aa71f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:27:01Z |
| publishDate | 2021 |
| publisher | Electronic Journal of Combinatorics |
| record_format | dspace |
| spelling | oxford-uuid:56ff0042-85a1-488e-a83b-5571a00aa71f2022-03-26T16:53:53ZOn the maximum number of integer colourings with forbidden monochromatic sumsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:56ff0042-85a1-488e-a83b-5571a00aa71fEnglishSymplectic ElementsElectronic Journal of Combinatorics2021Liu, HSharifzadeh, MStaden, KLet f ( n , r ) denote the maximum number of colourings of A ⊆ { 1 , … , n } with r colours such that each colour class is sum-free. Here, a sum is a subset { x , y , z } such that x + y = z . We show that f ( n , 2 ) = 2 ⌈ n / 2 ⌉ , and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of f ( n , r ) for r ⩽ 5 . Similar results were obtained by Hán and Jiménez in the setting of finite abelian groups. |
| spellingShingle | Liu, H Sharifzadeh, M Staden, K On the maximum number of integer colourings with forbidden monochromatic sums |
| title | On the maximum number of integer colourings with forbidden monochromatic sums |
| title_full | On the maximum number of integer colourings with forbidden monochromatic sums |
| title_fullStr | On the maximum number of integer colourings with forbidden monochromatic sums |
| title_full_unstemmed | On the maximum number of integer colourings with forbidden monochromatic sums |
| title_short | On the maximum number of integer colourings with forbidden monochromatic sums |
| title_sort | on the maximum number of integer colourings with forbidden monochromatic sums |
| work_keys_str_mv | AT liuh onthemaximumnumberofintegercolouringswithforbiddenmonochromaticsums AT sharifzadehm onthemaximumnumberofintegercolouringswithforbiddenmonochromaticsums AT stadenk onthemaximumnumberofintegercolouringswithforbiddenmonochromaticsums |