The largest $(k,\ell )$-sum-free subsets
<p>Let M(2,1)(N) be the infimum of the largest sum-free subset of any set of N positive integers. An old conjecture in additive combinatorics asserts that there is a constant c = c(2, 1) and a function ω(N) → ∞ as N → ∞, such that cN + ω(N) < M(2,1)(N) < (c + o(1))N. The constant c(2, 1)...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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American Mathematical Society
2021
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| _version_ | 1826277665120714752 |
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| author | Jing, Y Wu, S |
| author_facet | Jing, Y Wu, S |
| author_sort | Jing, Y |
| collection | OXFORD |
| description | <p>Let M(2,1)(N) be the infimum of the largest sum-free subset of
any set of N positive integers. An old conjecture in additive combinatorics
asserts that there is a constant c = c(2, 1) and a function ω(N) → ∞ as
N → ∞, such that cN + ω(N) < M(2,1)(N) < (c + o(1))N. The constant
c(2, 1) is determined by Eberhard, Green, and Manners, while the existence of
ω(N) is still wide open.</p>
<p>In this paper, we study the analogous conjecture on (k,ℓ)-sum-free sets
and restricted (k,ℓ)-sum-free sets. We determine the constant c(k,ℓ) for every
(k,ℓ)-sum-free sets, and confirm the conjecture for infinitely many (k,ℓ).</p> |
| first_indexed | 2024-03-06T23:32:20Z |
| format | Journal article |
| id | oxford-uuid:6c7b72f8-aef5-424f-9fdc-1aad8d17bdcf |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T23:32:20Z |
| publishDate | 2021 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:6c7b72f8-aef5-424f-9fdc-1aad8d17bdcf2022-03-26T19:11:03ZThe largest $(k,\ell )$-sum-free subsetsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:6c7b72f8-aef5-424f-9fdc-1aad8d17bdcfEnglishSymplectic ElementsAmerican Mathematical Society2021Jing, YWu, S<p>Let M(2,1)(N) be the infimum of the largest sum-free subset of any set of N positive integers. An old conjecture in additive combinatorics asserts that there is a constant c = c(2, 1) and a function ω(N) → ∞ as N → ∞, such that cN + ω(N) < M(2,1)(N) < (c + o(1))N. The constant c(2, 1) is determined by Eberhard, Green, and Manners, while the existence of ω(N) is still wide open.</p> <p>In this paper, we study the analogous conjecture on (k,ℓ)-sum-free sets and restricted (k,ℓ)-sum-free sets. We determine the constant c(k,ℓ) for every (k,ℓ)-sum-free sets, and confirm the conjecture for infinitely many (k,ℓ).</p> |
| spellingShingle | Jing, Y Wu, S The largest $(k,\ell )$-sum-free subsets |
| title | The largest $(k,\ell )$-sum-free subsets |
| title_full | The largest $(k,\ell )$-sum-free subsets |
| title_fullStr | The largest $(k,\ell )$-sum-free subsets |
| title_full_unstemmed | The largest $(k,\ell )$-sum-free subsets |
| title_short | The largest $(k,\ell )$-sum-free subsets |
| title_sort | largest k ell sum free subsets |
| work_keys_str_mv | AT jingy thelargestkellsumfreesubsets AT wus thelargestkellsumfreesubsets AT jingy largestkellsumfreesubsets AT wus largestkellsumfreesubsets |