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Contents
  1. C78-7: The extremal number of stable-marriage matchings
  2. 1) FORMAL RESTATEMENT
  3. 2) QUICK LITERATURE/CONTEXT CHECK
  4. 3) ATTACK PLAN
  5. 4) WORK
  6. 5) VERIFICATION
  7. 6) FINAL
  8. Reproducibility and scope.
  9. References

Research attempt and adversarial verification

9 October 2026 Original-target status: UNRESOLVED

C78-7: The extremal number of stable-marriage matchings

Input: C78-7.tex.  Tools: web browsing [YES]; Python computation [YES].

1) FORMAL RESTATEMENT

For every integer $n\geq1$, let $U=\{u_1,\ldots,u_n\}$ and $W=\{w_1,\ldots,w_n\}$ be disjoint. Each agent has a strict total order on the other side. A perfect matching is a bijection $M:U\to W$. It is stable when, for every $u\in U$ and $w\ne M(u)$, at least one of $u,w$ prefers its partner in $M$ to the other. Write $s(I)$ for the number of stable matchings of the preference profile $I$ and \[ x_n=\max_I s(I),\qquad \gamma=\limsup_{n\to\infty}x_n^{1/n}. \] The task is to determine $x_n$, or sharp asymptotics including $\gamma$ and, ideally, subexponential factors. It is an extremal determination problem, not the assertion that some given formula is true. Merely displaying an instance with few or many matchings does not disprove the question.

We set $x_0=1$ for block arguments; this convention does not alter the input. Preferences are complete, ties are forbidden, and labeled matchings are counted without identifying isomorphic objects. Important stress points are cross-block matches, strictness of preferences, and the distinction between a limit and a limsup.

2) QUICK LITERATURE/CONTEXT CHECK

The survey [matching-survey] explicitly asks the extremal counting question. Palmer and Pálvölgyi prove the upper bound $x_n\leq 3.55^{n+O(1)}$ in Theorem 1 of [palmer]; that paper also records an exponential lower base approximately $2.28$. These are literature bounds, not new bounds established below. The targeted search did not yield a verified determination of the exact base.

3) ATTACK PLAN

Proof track. First establish exact factorization for independent blocks. Next exploit the preference structure of the symmetric difference of two stable matchings. Finally test whether the small-instance structure can constrain the asymptotic base.

Construction track. Search all profiles of sizes $1,2,3$; construct independent copies of an extremal small block; then examine why such products do not by themselves bound more strongly interacting instances.

Tools and purpose. Deferred acceptance guarantees a nonempty stable set; alternating cycles compare stable matchings; permutation parity rules out an extremal three-agent chain; block products amplify constructions; logarithmic superadditivity identifies the growth limit; exhaustive permutation enumeration checks the small cases; rotation-poset/entropy methods are possible tools for a future upper bound, but are not silently invoked in the proofs below.

4) WORK

Lemma (Existence and exact block factorization): Every complete strict instance has a stable matching. If two instances are placed in disjoint blocks and every agent ranks all members of its own block ahead of all outsiders, then the stable matchings of the combined instance are exactly the unions of one stable matching from each block. Consequently \[ x_{n+m}\geq x_nx_m\qquad(n,m\geq0). \]
Proof: For existence, let unmatched men propose in preference order and let each woman retain her preferred proposal so far. There are at most $n^2$ proposals. An unmatched man cannot exhaust his list: otherwise every woman has received a proposal and retains a man, which would leave no man unmatched. Thus the procedure ends with a perfect matching. A man who prefers another woman to his final partner proposed to her earlier; she rejected him, or later replaced him, and her retained partner can only improve. Hence that pair cannot block.

For factorization, suppose a perfect matching uses an external partner for some man in a block of size $n$. The number of internally matched men equals the number of internally matched women. Therefore a woman in the same block also has an external partner. These two agents prefer one another to their external partners, so they block. Thus any stable matching is internal to every block. Its restriction to a block must be stable, or an internal blocking pair would block the whole matching. Conversely, a union of stable block matchings has no internal blocking pair, and neither member of a cross-block pair prefers the outsider to its own internal partner. Multiplying the numbers of choices proves the assertion. ∎

This lemma supplies a lower-bound construction and the only supermultiplicativity used below; it does not assert that extremizers themselves decompose into blocks.

Theorem (The growth limit exists): Using the exponential upper bound stated in the literature check above, \[ \lim_{n\to\infty}x_n^{1/n}=\sup_{k\geq1}x_k^{1/k}=\gamma<\infty, \qquad x_n\leq\gamma^n\quad(n\geq1). \]
Proof: Fix $k\geq1$ and write $n=qk+r$, where $0\leq r<k$. The previous lemma and $x_r\geq1$ give $x_n\geq x_k^q$. Since $q/n\to1/k$, $\liminf_n x_n^{1/n}\geq x_k^{1/k}$. Taking the supremum over $k$ proves that the liminf is at least the supremum of the entire sequence. The reverse inequality for the limsup follows from the definition of a supremum. The cited exponential upper bound makes this supremum finite. Each term is bounded by the supremum, which proves the last inequality. ∎
Lemma (Preference orientation on an alternating cycle): Let $M,N$ be two stable matchings. On any single alternating cycle in their symmetric difference, all men prefer $M$ to $N$, or all prefer $N$ to $M$.
Proof: Index the cycle so that $M(u_i)=w_i$ and $N(u_i)=w_{i+1}$, cyclically. Suppose $u_i$ prefers $w_i$ to $w_{i+1}$. Stability of $N$ forces $w_i$ to prefer $u_{i-1}$ to $u_i$. Stability of $M$ then forces $u_{i-1}$ to prefer $w_{i-1}$ to $w_i$. Iterating this implication around the cycle makes every man prefer $M$. If the first man instead prefers $N$, interchange the two matchings. Strict preferences ensure that these are the only cases. ∎
Theorem (Exact small values, with a human-checkable upper bound): \[ x_1=1,\qquad x_2=2,\qquad x_3=3. \]
Proof: The first equality follows from there being only one bijection. For $n=2$, the upper bound is $2!=2$. The instance \[ u_1:w_1\succ w_2,\quad u_2:w_2\succ w_1,\qquad w_1:u_2\succ u_1,\quad w_2:u_1\succ u_2 \] has both bijections stable: in the identity matching the men have their first choices, and in the other matching the women have their first choices.

For $n=3$, any two different permutations differ on a single alternating cycle, of length two or three. The preceding lemma therefore makes any two stable matchings comparable in every man's preference order. These comparisons form a linear order: transitivity follows from the transitivity of each individual preference order.

Suppose four stable matchings existed. Order four of them as $M^0\succ M^1\succ M^2\succ M^3$, from the men's perspective. Every consecutive change moves at least two men, so there are at least six partner changes. Each of the three men can change partners at most twice while strictly worsening among three women. There are consequently exactly six changes: each step moves exactly two men, and every man moves exactly twice. Each step is a transposition of partners, so the permutation taking $M^0$ to $M^3$ is odd. On the other hand, each man has visited three distinct partners, so this permutation has no fixed point. A fixed-point-free permutation of three objects is a three-cycle and is even. This contradiction proves $x_3\leq3$.

For the matching lower bound, use the lists \[ \begin{array}{c|c@{\qquad}c|c} u_1&w_1\succ w_2\succ w_3&w_1&u_2\succ u_3\succ u_1\\ u_2&w_2\succ w_3\succ w_1&w_2&u_3\succ u_1\succ u_2\\ u_3&w_3\succ w_1\succ w_2&w_3&u_1\succ u_2\succ u_3 \end{array} \] and the matchings $(w_1,w_2,w_3)$, $(w_2,w_3,w_1)$, and $(w_3,w_1,w_2)$, where coordinates list partners of $u_1,u_2,u_3$. The first gives all men their first choices and the last gives all women their first choices. In the middle matching, a man's only preferred woman is his first choice. That woman prefers her middle-matching partner to him, as is verified in each of the three displayed lists. Hence all three matchings are stable. ∎

In particular, independent three-agent blocks give $x_{3q+r}\geq3^q x_r$ for $r=0,1,2$. This elementary lower bound is weaker than the literature lower bound; it is included because its proof is complete here.

5) VERIFICATION

The exhaustive check in verify.py considers exactly $(n!)^{2n}$ preference profiles and all $n!$ matchings for each profile. It constructs inverse rankings and rejects a matching precisely when both blocking inequalities hold. The reported histograms are \[ \begin{array}{c|r|rrr} n&\text{profiles}&s(I)=1&s(I)=2&s(I)=3\\\hline 1&1&1&0&0\\ 2&16&14&2&0\\ 3&46656&34080&11484&1092 \end{array} \] Thus the computations independently check, rather than replace, the analytical small-case proof. A minimal pseudocode description is 
\begin{lstlisting} for all 2*n lists, each a permutation of range(n):     count = 0     for every bijection M:         if no pair prefers one another to their partners:             count += 1     update histogram and maximum \end{lstlisting}
The alternating-cycle upper bound is deliberately limited to $n=3$: for $n\geq4$ the symmetric difference can have several cycles with different orientations. That is the exact point at which this proof stops extending. Complete lists are also essential to the cross-block counting argument as stated. No claim about random preference instances, counting complexity, or a sharp base follows from these finite checks.

\Needspace{8\baselineskip}

6) FINAL

LABEL: UNRESOLVED.

Strongest checked partial result. Exact factorization under independent blocks; the limit formula for $\gamma$; and a complete proof of $x_1=1,x_2=2,x_3=3$. The record numerical bases in the literature check above are imported literature bounds.

Exact first gap. There is no matching universal upper and constructive lower exponential bound determining $\gamma$ in this work.

Next three moves.

  1. Derive a restriction on alternating-cycle orientations that survives when several cycles coexist, rather than assuming the three-agent linear order.
  2. Search for interacting blocks whose stable-count-per-agent exceeds that of their components, keeping complete preference-list certificates.
  3. Relate those restrictions to a sharper count of admissible rotation-poset downsets; an arbitrary-poset bound would need its representation hypotheses checked.
Counterexample perspective. There is no specified numerical conjecture to falsify. A construction beating a proposed exponential upper bound would need an infinite amplified family, not just a mislabeled small-instance maximum.

Reproducibility and scope.
This is one of eleven companion reports. The bundle includes verify.py and its executed results. Finite experiments supplement the proofs; they are not proofs of asymptotic claims. Literature coverage is targeted, and no novelty or formal proof-assistant certification is claimed.

References