A309370 comments

[Konstantin Knop]

Hello, SeqFans!

I have a question about this comment in A309370 text

"Define addition on {0,1}^n componentwise (ordinary addition, not
addition modulo 2, so the result lies in {0,1,2}^n, not necessarily
{0,1}^n). We say a subset of {0,1}^n is Sidon iff the only solutions
to a+b = c+d, with a,b,c,d in the set, are the trivial ones: a=c, b=d
or a=d, b=c.

a(7) >= 23, a(8) >= 32, a(9) >= 45, a(10) >= 63, a(11)>=87,
a(12)>=120, a(13)>=169, a(14)>=237, a(15) >= 334, a(16) >= 472, a(17)
>= 662, a(18) >= 864.

Conjecture: a(n) is asymptotic to 2^(n/2+1).

a(7) >= 24. - _Christian Sievers_, Sep 17 2025"

-------
Where and how can I see examples of sets of vectors [0,1]^n that form
the sets specified in these inequalities? For example, 24 vectors for
n=7?

KK, http://knop.website

[Michel Marcus]

You could send end a mail to Christian Sievers.

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[Charles Greathouse]

It would be great for those to be added to the Examples. They could probably be represented as binary strings:

0 {(empty string)}

1 {0, 1}

2 {00, 01, 10}

...

I'd also love to see the Cohen, Litsyn, & Zémor upper bound. (Failing that, we could add the trivial upper bound a(n) <= 2^n.) Do we have any lower bounds stronger than a(n) > a(n-1)?

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[Victor Miller]

Here’s a link to the Cohen/Zemor/Litsyn result: 

https://web.archive.org/web/20240415153501/https://core.ac.uk/download/pdf/82159089.pdf

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[Victor Miller]

Here’s another more recent comprehensive reference: 

https://arxiv.org/pdf/1809.07364

[Christian Sievers]

I put one example for a(7)>=24 in the discussion that you see when you click on "history".

If it's not in the links that were given, I can also give an example confirming a(10)>=63.

[D. S. McNeil]

I wasn't able to improve on a(7)-a(9), but I was able to grind out some improvements for n>10:

  n  best |S|
---------------
  2         3
  3         5
  4         7
  5        12
  6        15
  7        24
  8        32
  9        45
 10        65
 11        92
 12       130
 13       179
 14       253
 15       338
 16       476
 17       671
 18       942

The witnesses are inconveniently long for the comments, though.

Doug

[Charles Greathouse]

That’s great, Doug! Maybe an a-file?

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[D. S. McNeil]

Unfortunately they're only one-sided bounds, or I would. :-) 

I'll toss in a comment, though, was just waiting to let a few improve and then for the open edits to go in.

Doug

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