co.combinatorics – Powerful existence theorems with mild conditions: more recent examples

I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that a minimum regular pattern always exists. By mild conditions I mean short, easy, broad conditions. By simple conditions I mean not requiring advanced mathematical education. The conditions and the statement should be accessible to undergraduate mathematics/science students.

I am interested mostly in low-dimensional examples which allow an easy graphical representation.

I have some obvious examples in mind (given below), but they are rather classical results that were established between 1900 and 1950, roughly speaking.
I would be interested to see examples that are more recent.

Classical examples I have in mind

(1) Lemma of Sperner and Brouwer Fixed Point Theorem (for $n=2$)

(2) Lemma of Tucker and Borsuk-Ulam Theorem (for $n=2$)

(3) Ramsey’s Theorem (for the simplest case of 6 edges)

(4) Wagner’s Theorem about Planar Graphs

I would be grateful if you could point me to more recent examples.

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algebraic geometry ag. – Suitability of reductive group actions in mild varieties

Suppose that $ G $ it is a reductive algebraic group that acts on a mild variety $ X $, and that the action has finite stabilizers. When is the action of $ G $ in $ X $ appropriate? What is an example where the action is not adequate?

I am aware of a similar statement that is Proposition 0.8 in the Mumford GIT, which says that the action is appropriate if the geometric quotient $ phi: X a X / G $ exists and $ phi $ It is similar I would like to know in what situations one can assume that the action is adequate without this assumption.

related varieties – Tangent package for a mild algebraic variety

I was discussing with a friend how to define the tangent package for a mild variety, since this is very natural in the multiple environment and we could not find references discussing in detail.

For example, yes $ X subset mathbb {A} ^ n $ it is a related algebraic variety we can define the tangent package $ TX subset mathbb {A} ^ {2n} $where first $ n $ coordinates respect for the equations to $ X $ and the other the equations for tangent spaces: $ sum_ {i = 1} ^ n { frac { partial f} { partial x_i}} (x) y_i = 0 $. This implies that $ TX $ It is an algebraic variety, but my problem is: is this a vector pack? The problem is that I don't see how it will be locally trivial. Zariski's open sets are very large and will impose a lot of rigidity; Nor is there an implicit function theorem, so we can get trivializations to $ TX $ as in the complex analytical point of view.

If this is locally trivial, how can it be shown? Otherwise, can you easily find an example?

at.algebraic topology – Are all varieties in $ mathbb R ^ n $ homeomorphic to a mild variety?

My question is whether all the multiples that can be embedded in $ mathbb R ^ n $ Are they homeomorphic to a mild variety?

I know that each smooth manifold can be triangulated, which I think is the result of Whitehead and I believe that each manifold in $ mathbb R ^ n $ It can be triangulated, so this gives credibility, I think. (If the multiple of dimension 4 E8 can be embedded in $ mathbb R ^ n $ So we have a counterexample but I'm not sure if it can be. All the collectors of dimension up to 3 can be triangulated).

I appreciate if this is a basic question, but it doesn't seem to be explicitly explained anywhere. Most textbooks define a variety and then a mild variety, but do not say in which cases these concepts can be equivalent.

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