G-types

Math in View

Opening the workspace

H12Parity classes

ρ ≤ σ

$\rho: (2,1)\qquad\sigma: (3,2)$

C is an actual subset of B with the same G-type as A.

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0–999Finite window in ℕ

G-types

Representatives

Click a number to toggle it in the active set, or edit the sets below. Empty selections have no KPT type.

Random finite set

Sample without replacement from the configured window. Change the seed for another sample.

Number of selected pointsSeed

Window into ℕ

The default is 1,000 points. You choose its starting point and size; the sets remain selected outside the window.

Another member of ρ

Apply one of three computable members of the entire group to A. The purple squares mark the resulting finite set.

H12 · FINITE-SET ORBITS

Parity classes

Current representatives

$\rho=[\{0,3,8\}]_G$ , invariant $(2,1)$ .

$\sigma=[\{2,4,6,7,9\}]_G$ , invariant $(3,2)$ .

$\rho\leq\sigma$ : choose $C=\{2,4,7\}\subseteq B$ . Then $[C]_G=\rho$ .

Match even points to even points and odd points to odd points; extend independently in the two classes.

$\sigma\nleq\rho$ . A has more points than B. A permutation preserves cardinality.

Finite matching that extends to an element of G

0 ↦ 2; 3 ↦ 7; 8 ↦ 4

Curves on the canvas show at most the first 40 matched pairs whose endpoints lie in the window.

Definition, invariant, and proof

Definition. $G$ consists of permutations preserving the even numbers and the odd numbers separately. These are named classes, so swapping the two classes is forbidden.

Invariant. The type is $(e,o)$ , the numbers of even and odd points. The order is coordinatewise:

(e,o)\leq(e',o')\quad\Longleftrightarrow\quad e\leq e'\ \text{and}\ o\leq o'.

Why it works. Independently choose $e$ even and $o$ odd points in $B$ . Match within each class and extend the two finite bijections on the infinite class complements.

Topology and significance. Preservation of the named unary predicate is closed: a failure has a one-point witness. The group is not open, since outside any finite fixed set one can exchange an even and an odd point. The type order is the nonzero part of $\mathbb N^2$ with coordinatewise order.

Explore. The default sets have types $(2,1)$ and $(3,2)$ even though $A$ is not literally a subset of $B$ . Also compare $(2,0)$ with $(1,2)$ .

G-types, the partial order, and KPT

AI-assisted mathematical commentary, written for the interactive exploration. The definitions follow the discussion immediately before Proposition 4.3 in Kechris–Pestov–Todorcevic.

For a subgroup $G\leq S_\infty$ and a nonempty finite set $A\subseteq\mathbb N$ , its type is

[A]_G=G\cdot A=\{g[A]:g\in G\}.

Each element of this orbit is itself a finite subset. Selecting points specifies one representative, not the entire type. A few displayed representatives are samples of an orbit, not an enumeration of it.

For $\rho=[A]_G$ and $\sigma=[B]_G$ , the KPT relation is

\rho\leq\sigma\quad\Longleftrightarrow\quad \exists C\subseteq B\;[C]_G=[A]_G.

Equivalently, some $g\in G$ sends $A$ into $B$ . The app displays $C=g[A]$ and a finite matching $A\to C$ . The extension proofs below explain why this matching belongs to a full group element. The curves show only the finite matching.

Why this is independent of representatives

If $g[A]\subseteq B$ , then for any $u,v\in G$ , $vgu^{-1}[u[A]]\subseteq v[B]$ . Thus the same comparison works for every choice of representatives. In particular, one witness in $B$ implies a witness in every member of $\sigma$ .

Why it is a partial order

Identity gives reflexivity. Composing embeddings gives transitivity. If $\rho\leq\sigma$ and $\sigma\leq\rho$ , their finite representatives have equal sizes. A subset of $B$ of size $|B|$ is $B$ itself, so the types are equal. This proves antisymmetry.

Cardinality is a rank: strict comparison increases size. Every cover increases size by exactly one, because a gap in size can be filled by adjoining one point inside a larger representative. The diagram therefore draws only comparable pairs whose sizes differ by one. Increasing rank goes upward. Paths give further comparisons; missing paths in the displayed finite portion do not prove incomparability.

Reading a finite window

The line and winding grid both label ordinary natural numbers; winding through rows introduces no new algebraic structure. Moving or enlarging the window never changes a selected set. Some selected points and witnesses may lie outside it. Zooming reveals more background labels; it does not change the set or type.

The order diagram includes the current types and a stated finite catalogue. For the pair-flip and translation groups this catalogue is only a portion even at the shown ranks. Larger selected types can be separated by ranks absent from the catalogue.

For rational examples, $n$ denotes the rational $q_n$ in the fixed enumeration by increasing $|a|+b$ , then numerator $a$ , for reduced $a/b$ with $b>0$ . The ordinary grid is not in rational order. Inspect the rational coordinates or ordered colour word in Mathematics.

Relation to Proposition 4.3

These seven groups are closed in the pointwise-convergence topology. Closedness supplies the setting of Proposition 4.3; it does not by itself imply its Ramsey conclusion. That proposition also concerns finite-set stabilizers and colourings of members of a type. The present exploration prepares the types and their order; it does not test or assert the colouring property.

Computation and limits

The seven invariants below classify finite-set orbits exactly. Comparison algorithms decide the order and return a witness when completed. A rational-decoding or translation-search work limit is reported as undetermined, never as an obstruction. Random sets are reproducible pseudorandom samples without replacement from the configured finite window. This is no uniform distribution on all finite subsets of $\mathbb N$ .