Karolyi's theorem for finite groups and its extensions Suppose that $\mathbb A = (A, +)$ is a (possibly non-commutative) group, and denote by $p(\mathbb A)$ the minimum of $|S|$ as $S$ ranges in the set of non-trivial subgroups of $\mathbb A$, with the convention of taking $p(\mathbb A) := 1$ if $\mathbb A$ is trivial. Then, pick non-empty subsets $X$ and $Y$ of $A$. The following result is part of the folklore (in fact, a straightforward application of Kneser's theorem): 


Theorem 1. If $\mathbb A$ is commutative then $|X+Y| \ge \min(p(\mathbb A), |X| + |Y| - 1).$


(This boils down to the (classical) Cauchy-Davenport theorem when $\mathbb A$ is a cyclic group of prime order.) The same result is known to hold if "commutative" in the previous statement is replaced with "finite"; this was first proved by G. Károlyi in 2005 [1], based on the Feit-Thompson theorem by reduction to the case of finite solvable groups, and recently re-proved by Ruzsa (see the comments below) as a consequence of a stronger result, which gets even rid of the assumption of finiteness. I.e., Ruzsa proves the following:


Theorem 2. $|X+Y| \ge \min(p(\mathbb A), |X| + |Y| - 1)$ no matter if $\mathbb A$ is finite or infinite, commutative or not.


But Ruzsa's result is something new, so it is plausible that Theorem 2 has been presented as a "conjecture" for a while. Then, my first question is:


Q1. A similar "conjecture" should have at least one father/mother. What is his/her name? To wit, is there any paper, book, etc. where it has been first stated explicitly? For what I can say, this is the case neither with [1] nor with [2] (where Károlyi gives a self-contained proof of Theorem 2 for the abelian case that doesn't even refer to Kneser's theorem).


Now, Theorem 2 can be restated in much more general terms by assuming that $\mathbb A$ is a unital magma (instead of a group) and replacing "subgroups" in the above with "submagmas". So my next question is:


Q2. What is known about the general "conjecture"? That is, are there partial results related to (classes of) magmas which do not embed into a group?


For any purpose it may serve, let me mention that I've something in these lines for the case when $\mathbb A$ is a cancellative monoid, $X \cap \mathbb A^\times$ is non-empty, and the smallest submonoid of $\mathbb A$ containing $X$ is commutative (or dually with $Y$ in place of $X$). Also, I believe I've a proof in the case when $\mathbb A$ is a cancellative monoid, either commutative or not, and $(X+Y) \cap \mathbb A^\times$ is non-empty. But I dislike much the assumption on the units (somehow, it doesn't look very "natural"), so any insight that could help to get rid of it would be greatly appreciated. In particular, I'd like to hear of alternative (or possibly more general) results.
Update (12/06/2013). It was observed below, in the comments to Gerhard Paseman's answer, that there exist commutative non-associative magmas with arbitrarily large subsets for which Q2 is answered in the negative. So it seems natural to ask the following:


Q3. Does the general "conjecture" hold if $\mathbb A$ is associative and/or cancellative?


Bibliography.
[1] G. Károlyi, The Cauchy-Davenport theorem in group extensions, L'Enseignement Mathématique 51 (2005), 239-254.
[2] -----------, A compactness argument in the additive theory and the polynomial method, Discrete Math. 302 Nos 1-3 (2005), 124-144.
 A: This is weakly related to an answer to your second question.
Also it is fragmentary, and possibly some details are misremembered.
It relates to some of the first original mathematics I did over 20 years
ago.
There is a result of Vadim Murskii, which goes by the tagline "Almost
All Finite Algebras Are Finitely Based".  Fix a similarity type which
includes at least one function of arity two. (There may be a way to
handle structures which have only unary functions, but I don't recall
it.)  Consider all structures of that type on a (labeled) set of n elements.
Since it has a binary operation, there are at least n^(n^2) such structures,
and likely many more.  If one looks at a structure A, one can note a
property that the structure has.  A is finitely based if the equational theory
of the variety generated by A is equivalent to a theory which is generated
by finitely many axioms in equational logic.  We say almost all algebras are
finitely based if, as n gets large, the proportion of finitely based labeled algebras
on n elements compared to all such algebras on n elements goes to 1.
There are a couple of ways to prove such a result, and I studied them
and gave a talk on them as a graduate student.  I went from notes
of R. Quackenbush, as well as attempting to read Murskii's papers from
the 1960's and 1970's.  One key idea was to reduce the problem to
looking at magmas, and another was to look at certain classes of
magmas which were able to be algebraically classified and shown to
be provably finitely based.  Looking at idempotent magmas was one
of the interesting features of the analysis.
Indeed, part of the argument was to show that if a magma had one
of eight or ten special properties, then something nice occurred,
generally you could show it was part of a negligible fraction of
such algebras, or it was related to a finitely based magma, or something
else that occurred.  
To tie this back to your question, one of the properties talked about
a deficient magma, which is a special form of your property.  If A is
a magma on n > 3 elements, it is deficient if there is a subset B of at
least 3 and fewer than n elements, such that the cardinality of B is
at least the cardinality of B*B, where I am using * for the magma
operation.  This is more general than having a not too small subalgebra.
Also, being deficient turns out to be relatively rare.  (I wondered why
B having 2 elements was excluded.  I discovered those magmas are
not rare, and I also discovered a simpler proof that almost all magmas
were not deficient.)
If you were to find the notes of Quackenbush, or use a citation index
starting with Murskii, or do a web search of "almost all" finite algebras,
you might find some papers like Ralph Freese's on probability in algebra,
or work of Sapir and McNulty on the finite basis problem.  Hopefully
though, you will find something that fully addresses your question 2.
Gerhard "Ask Me About 2-Deficient Groupoids" Paseman, 2013.06.11
