A Note on the Erdös Distinct Subset Sums Problem

2021 ◽  
Vol 35 (1) ◽  
pp. 322-324
Author(s):  
Quentin Dubroff ◽  
Jacob Fox ◽  
Max Wenqiang Xu
Keyword(s):  
Distinct Subset ◽  
Subset Sums

scholarly journals Integer sets with distinct subset-sums

1988 ◽  
Vol 50 (181) ◽  
pp. 297-297 ◽  
Author(s):  
W. F. Lunnon
Keyword(s):  
Distinct Subset ◽  
Subset Sums

scholarly journals A construction for sets of integers with distinct subset sums

10.37236/1341 ◽  
1997 ◽  
Vol 5 (1) ◽  
Author(s):  
Tom Bohman
Keyword(s):  
Upper Bound ◽  
Distinct Subset ◽  
Subset Sums ◽  
Positive Integers

A set S of positive integers has distinct subset sums if there are $2^{|S|}$ distinct elements of the set $\left\{ \sum_{x \in X} x: X \subset S \right\} . $ Let $$f(n) = \min\{ \max S: |S|=n {\rm \hskip2mm and \hskip2mm} S {\rm \hskip2mm has \hskip2mm distinct \hskip2mm subset \hskip2mm sums}\}.$$ Erdős conjectured $ f(n) \ge c2^{n}$ for some constant c. We give a construction that yields $f(n) < 0.22002 \cdot 2^{n}$ for n sufficiently large. This now stands as the best known upper bound on $ f(n).$

scholarly journals The number of distinct subset sums of a finite set of vectors

1993 ◽  
Vol 63 (2) ◽  
pp. 234-256
Author(s):  
Ernest Brickell ◽  
Michael Saks
Keyword(s):  
Distinct Subset ◽  
Finite Set ◽  
Subset Sums

scholarly journals ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS

2021 ◽  
pp. 1-7
Author(s):  
SÁNDOR Z. KISS ◽  
VINH HUNG NGUYEN
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Distinct Subset ◽  
Subset Sums ◽  
Asymptotic Basis ◽  
Positive Integers

Abstract Let k and l be positive integers satisfying $k \ge 2, l \ge 1$ . A set $\mathcal {A}$ of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as the sum of k terms from $\mathcal {A}$ . About 35 years ago, P. Erdős asked: does there exist an asymptotic basis of order k where all the subset sums with at most l terms are pairwise distinct with the exception of a finite number of cases as long as $l \le k - 1$ ? We use probabilistic tools to prove the existence of an asymptotic basis of order $2k+1$ for which all the sums of at most k elements are pairwise distinct except for ‘small’ numbers.

Lung Tumors With Neuroendocrine Morphology: Essential Radiologic and Pathologic Features

2008 ◽  
Vol 132 (7) ◽  
pp. 1055-1061 ◽  
Author(s):  
Teri J. Franks ◽  
Jeffrey R. Galvin
Keyword(s):  
Neuroendocrine Tumors ◽  
Classification Systems ◽  
World Health ◽  
National Library ◽  
Molecular Features ◽  
Distinct Subset ◽  
Pathologic Features ◽  
The World ◽  
Health Organization

Abstract Context.—Tumors with neuroendocrine morphology are a distinct subset of lung neoplasms sharing characteristic histologic, immunohistochemical, ultrastructural, and molecular features. Objective.—To review the current histologic classification and the diagnostic criteria for the major categories of neuroendocrine tumors of the lung. Data Sources.—Published classification systems from the World Health Organization and pertinent peer-reviewed articles indexed in PubMed (National Library of Medicine) form the basis of this review. Conclusions.—Accurate classification of the neuroendocrine tumors of the lung requires knowledge of specific criteria separating the major categories, which is essential for determining prognosis and treatment.

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