TITLE: $\kappa$-homogeneous topological spaces
QUESTION [19 upvotes]: Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-homogeneous if

$|X| \geq \kappa$, and
whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$ and $\psi:A\to B$ is a bijective map, then there is a homeomorphism $\varphi: X\to X$ such that $\varphi|_A = \psi$.

Questions: Is it true that for $0<\alpha < \beta$ there is a space $X$ such that $|X|\geq \beta$, and $X$ is $\alpha$-homogeneous, but not $\beta$-homogeneous? Is there even such a space that is $T_2$? Also it would be nice to see an example for $\alpha=1, \beta=2$. And I was wondering whether there is a standard name for $\kappa$-homogeneous spaces. (Not all of these questions have to be answered for acceptance of answer.)

REPLY [7 votes]: The sort of space you describe is usually called strongly $\kappa$-homogeneous. If you google that phrase you will find some interesting results about these kinds of spaces (mostly concerning how this property relates to other homogeneity properties).
The earliest reference I could find to strongly $n$-homogeneous spaces (only finite values of $n$ are considered) is in a 1953 paper by C. E. Burgess (available here). 
Despite the fact that these kinds of spaces have been in the literature for well over half a century, it is unknown whether there is a topological space that is strongly $4$-homogeneous but not strongly $5$-homogeneous. (This is stated explicitly in this paper by Ancel and Bellamy from last year -- see the second paragraph of the second page.) As far as I can tell (although I don't have an authoritative reference), it is also unknown whether there is a strongly $n$-homogeneous space that is not also strongly $(n+1)$-homogeneous for any finite $n \geq 4$.
Therefore Joel's answer (along with some of Andreas's comments below it) constitutes the state-of-the-art knowledge on this question.