==== Table Seatings ====

The general problem is arranging a group of people into a number of tables so that everyone sits with everyone else. There are multiple versions for this.

  * The **strict** version is that all tables are the same size and that after the required number of rounds, everyone has shared a table with every other person exactly once
  * The **lower version** requires that each person shares a table with each other person at most once
  * The **upper version** requires that each person shares a table with each other person at least once

Some general thoughts. Each sitting defines a partition of the set of people, each part is one table.

====Strict====
A strict version is an affine plane. Example 25 people in 5 tables of 5, Point set is $Z_5 \times Z_5$, we take the tables to be the lines $L(a,b)={(x,y)| y=ax+b}$ and $L(a)={(a,y)| y in Z_5}$, the sitting is a parallel class (the 5 lines with the same slope a), so we have 6 sittings, $L(a,b)$ for $a=0,1,2,3,4$ and then the parallel class of $L(a)$. 

More generally we want a [[https://en.wikipedia.org/wiki/Block_design#Resolvable_2-designs|resolvable 2-design]]. We want resolvable $(v,k,1)$ designs. Resolvable is the parallelism. Maybe there is something like discrete hyperbolic geometry to deal with this, but we seem to have better combinatorial ideas below.


Versions include [[https://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem|Kirkman's Schoolgirl Problem]] (15 children walk in groups of 3, can they do this so that all pairs of girls walk together exactly once over a whole week) {[[https://oeis.org/search?q=schoolgirl&sort=&language=german&go=Suche|oeis]]}. This is the question as to resolvable $(v,3,1) 2\text{–designs}$, which if I understand it, exist $\iff v = 3 mod 6$.

For pairs, resolvable $(v,2,1) 2\text{–designs}$ exist only for $even v, v >= 4$.

Table size 4: Resolvable $(v,k,1)- 2\text{–designs}. 
[[https://www.semanticscholar.org/paper/The-spectrum-of-resolvable-designs-with-block-size-Vasiga-Furino/364fb4a75a38493ed2c86fa3589adfee6d2714f5|This paper]] says that necessary numerical conditions are sufficient except for a case that need not concern us.

==== Lower Version====

Just leave out some sittings on a strict version. Perhaps add a few nonexistant people to get a better distribution of people on tables with not all tables always full.

==== Upper Version ====

The "[[https://github.com/fpvandoorn/Dagstuhl-tables|Dagstuhl Happy Diner problem]]" is the version where everyone meets at least once. {[[https://oeis.org/A318240|oeis]]}

[[https://www.researchgate.net/publication/227715273_Equitable_resolvable_coverings|Equitable Resolvable coverings]] also seem to be a more strict form, where we can allow people to meet at most twice. {[[https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.10024|ref]]}

If we have people sitting at round tables and only interacting with their neighbours, then we have the more difficult [[https://en.wikipedia.org/wiki/Oberwolfach_problem|Oberwolfach Problem]]

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part of [[category mathematics]]