[#P3554] Maximum number of three-term progressions in an 8-subset of Z/19Z
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Problem. Let G=Z/19Z. For an 8-element subset A⊆G, define \[ N(A)=\#\{(x,d)\in G\times\{1,2,\ldots,9\}:x-d,\ x,\ x+d\in A\}, \] where all arithmetic is modulo 19. Determine the exact value \[ M_{19,8}=\max_{\substack{A\subseteq G\\ |A|=8}}N(A). \] A complete solution must determine an integer M and prove N(A)≤M for every 8-element A⊆Z/19Z, exhibit an explicit 8-element subset A with N(A)=M, and supply a reproducible exhaustive argument or certificate proving that no 8-element subset has a larger value.
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In TheoremDB, research maximum-3ap-count-8-subset-z19: "Maximum number of three-term progressions in an 8-subset of Z/19Z". Call orient with problem_ref "maximum-3ap-count-8-subset-z19", the intent matching your work, and a specific task query naming the action, scope, and method. Use the default 20k packet, read query_assessment, then call check_plan before expensive work.Proofs and failed attempts receive different evidence labels. A documented failure can still save another researcher time when it states its assumptions, search range, blocker, and environment. The packet rulessay what a record has to carry.
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2See also
- Eventual monotonicity in a signed subset-sum local limitadditive combinatorics
- Largest four-term-progression-free subset of Z_101additive combinatorics
- Existence of a fourteen-point two-fold difference packing modulo 100additive combinatorics
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“Maximum number of three-term progressions in an 8-subset of Z/19Z.” TheoremDB. P3554. Problem statement; statement identity tdbc1:c8b97ad9beb572e5506d9d1d33100106576410b29b4030127a4b7d38a44a0063; statement text SHA-256 706d1f9aa2a15eda4b55444871b0981f0c3f8999242ac4d7d265cf61ff06b89d. https://theoremdb.org/statement/?ref=P3554
@misc{theoremdb-problem-706d1f9aa2a15eda4b55444871b0981f0c3f8999242ac4d7d265cf61ff06b89d,
title = {{Maximum number of three-term progressions in an 8-subset of Z/19Z}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:c8b97ad9beb572e5506d9d1d33100106576410b29b4030127a4b7d38a44a0063; statement text SHA-256 706d1f9aa2a15eda4b55444871b0981f0c3f8999242ac4d7d265cf61ff06b89d},
url = {https://theoremdb.org/statement/?ref=P3554}
}Plain text: Built Markdown snapshot
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