Discussion Overview
The discussion revolves around finding a constant \( c > 0 \) such that for every finite set of non-zero integers \( B \), there exists a sum-free subset \( A \) of \( B \) with \( |A| \geq c|B| \). The conversation explores theoretical approaches, potential bounds for \( c \), and specific cases of integer sets.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants clarify that a sum-free set cannot contain zero, emphasizing the importance of specifying non-zero integers in the problem statement.
- One participant suggests simplifying the problem by considering only positive integers, proposing that the same constant \( c_p \) could apply to negative integers as well.
- Another participant notes that if all elements of \( B \) are odd, then \( A = B \) and \( c = 1 \) could be a solution.
- Several participants propose upper limits for \( c \), with one stating that if \( |B| = 3 \) and \( B \) is not sum-free, then \( c \leq 2/3 \), while another gives an example with \( B = \{1, 2\} \) leading to \( c \leq 1/2 \).
- One participant discusses the implications of adding elements with negative counterparts to a sum-free set, suggesting it could influence the value of \( c \).
- A later reply questions the validity of certain arguments regarding residues and modular arithmetic, indicating confusion and a need for clarification on the conditions for sum-free sets.
- Another participant introduces a potential lower limit for \( c \) based on the density of prime numbers and their divisibility properties within the set \( B \).
- One participant highlights that if \( |B| \) has a smallest prime divisor \( s \), then a subset \( A \) can be formed with elements having equal residues when divided by \( s \), leading to \( c = 1/s \).
- Concerns are raised about the possibility of \( s \) being arbitrarily large, which could prevent a universal constant from being established.
Areas of Agreement / Disagreement
Participants express various viewpoints on the existence and bounds of the constant \( c \). There is no consensus on a definitive value for \( c \), and multiple competing perspectives remain throughout the discussion.
Contextual Notes
Participants note limitations in their arguments, such as the dependence on the properties of integers in set \( B \) and the challenges in establishing universal constants due to varying conditions of the elements.