Largest n such that 1 to n can be Partitioned for no Element to be Sum of 2 Elements in Same Set

From ProofWiki
Jump to navigation Jump to search

Theorem

$44$ is the largest integer $n$ such that the set of integers from $1$ to $n$ can be partitioned into $4$ subsets such that no integer in any of these subsets is the sum of $2$ other integers in the same subset:

$\set {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}$
$\set {2, 7, 8, 18, 21, 24, 27, 33, 37, 38, 43}$
$\set {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}$
$\set {9, 10, 11, 12, 14, 16, 29, 31, 34, 35, 36}$


Proof


This theorem requires a proof.
In particular: Outline: find all possible solutions of partitioning $44$ into sum-free sets, then show that $45$ cannot be inserted in any of the sets. The search is, of course, done by computer. Note that $x$ and $2 x$ cannot be in the same set.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Largest_n_such_that_1_to_n_can_be_Partitioned_for_no_Element_to_be_Sum_of_2_Elements_in_Same_Set&oldid=531500"