In the above comment, I am assuming the minimum distance between vertices restriction is still in place.

I thought about that too. The problem is that the growth of degrees can be super quadratic. Also, you have to allocate circular arc "ahead of time" to avoid edge crossing, or else limit the arcs severely.

You might consider reading Ayn Rand's "Philosophy: Who Needs It?" (Her answer in brief: everyone. I imagine some of her tenets will aid the original poster. While it is good to have apparent utility to show an outsider, it should be remembered that utility is not always approached on strictly utilitarian paths. Emphasizing exploring the fringes of the known as well as the unknown is also important.)

Thanks for accepting the answer. I would still like to know more of the motivation and the corollary, even if it did not turn out to be much more interesting.

Also, the Vedanta are based on some old discoveries. You might check with scholars of ancient Indian history.

It seems clear to me that you will a) either find second or third hand evidence, e.g. some 12th century Arab scholar talking about a writing a couple of millenia old, or b) you will have to glean inferences from artifacts of that age. I would choose b) and start with whatever tax records there were from that era, or whatever records used algebra in whatever form. Good luck.

Because there are skeptics everywhere. Some of them came together in groups and formed mathematics departments. Others formed boards of education. I'm sure you've encountered similar groups. (No, I'm not from Missouri.)

If I were developing the subject (and if I could remember whether down sets were called filters or ideals, let's say filters), then I would call such maps filter maps. I am not a poset person and currently don't talk in a knowing way to poset people, so consider this an opinion.

Does diagonal matrix mean constant diagonal? Or are different elements allowed on the diagonal?

This "feels" like a topological condition, like a subbase maps to a subbase, a sort of subopen map. Have you considered this perspective?

Being on the wrong side of the paywall, and conflating the set D with its cardinality, I don't know if Davis looks at the associated equation $D^2 - D = \lambda(n^2 - 1)$ in the cited paper. Does he? Does anyone else in the combinatorial design literature talk about allowed values for $\lambda$?

Ah yes. . Slightly different unsolved problem from Joseph's, are there any points at rational distances from tthe 4 vertices of a unit square? Apologies for the conflation.

Indeed, a couning argument might show something mildly stronger: a non bipartite graph which is triangle free has either a minimum degree less than 2n/5, or the graph is 2n/5-regular. This could well be part of an undergraduate graph theory course.

Given the elementary nature of the analysis in this answer, I would like to see the corollary mentioned in the question and how it is more interesting than the supposedly weaker alternative.

I think an open problem is whether the vertices of a square are a maximal such set. Here is a throwaway conjecture: for all n excepting those less than 4 and n=6, the vertices of a regular n-gon are a maximal example.

It might be useful to consider two vertices of minimum degree joined by an edge. If the graph is triangle free that gets a bound close to what is suggested, but there is more lurking behind the example.

Perhaps instead of divide by p, perhaps a step is gcd with a large power of p.