Three Generalizations of Erdős Szekeres: $k$-Modal Subsequences

Erdős and Szekeres showed that given a permutation $p$ of $[n]$, and the sequence defined by \newline $(p(1), p(2), \ldots, p(n))$, there exists either a decreasing or increasing subsequence, not necessarily contiguous, of length at least $\sqrt{n}$. Fan Chung considered subsequences that can have at most one change of direction, i.e. an increasing and then decreasing subsequence, or a decreasing and then increasing subsequence. She called these unimodal subsequences, and showed there exists a unimodal subsequence of length at least $\sqrt{3n}$, up to some constants \cite{chung}. She conjectured that a permutation of $n$ contains a $k$-modal (at most $k$ changes in direction) subsequence of length at least $\sqrt{(2k+1)n}$ up to some constants. Zijian Xu proved this conjecture in 2024 \cite{xu}, and we will provide another substantially different proof using “sophisticated labeling arguments” instead of “underlying poset structures behind k-modal subsequences.” We also show that there exists an increasing first $k$-modal subsequence of length at least $\sqrt{2kn}$.

💡 Research Summary

The paper revisits the problem of finding long subsequences in a permutation that change direction only a limited number of times. The classical Erdős–Szekeres theorem guarantees a monotone (0‑modal) subsequence of length at least √n. Fan Chung later proved that a subsequence with at most one change of direction (a unimodal or 1‑modal subsequence) always has length at least √(3n) up to constant factors, and conjectured that a k‑modal subsequence (at most k direction changes) should have length on the order of √((2k+1)n). This conjecture was settled by Zijian Xu in 2024.

The present work offers a completely different proof of Xu’s result, based on a “sophisticated labeling” technique rather than the poset‑theoretic arguments used previously. The authors assign to each element a_i of a permutation two integer labels:

• x(a_i) = length of the longest increasing subsequence ending at a_i,
• y(a_i) = length of the longest decreasing (or k‑modal decreasing) subsequence beginning at a_i (or ending, depending on the theorem).

A key observation is that the map f : a_i ↦ (x(a_i), y(a_i)) is injective. The proof of injectivity uses only the elementary fact that if a_i < a_j then x(a_i) < x(a_j), and similarly for the y‑labels. Consequently the whole permutation can be embedded as a set of lattice points inside a region of the integer grid.

For the base case k = 1 the points must lie in the triangle defined by x + y ≤ N + 1, where N is the length of the longest unimodal subsequence. Counting lattice points gives n ≤ N(N+1)/2, which yields N ≥ √(2n) (up to constants). The authors then perform an induction on k. Assuming the statement holds for k, they define y(a_i) to be the length of the longest k‑modal decreasing subsequence starting at a_i and repeat the injectivity argument. The image now lies in the union of two regions: a smaller triangle where y < kN/(k+1) and a complementary rectangle. Estimating the number of lattice points in both regions gives n ≤ N²/(2(k+1)²), which translates to N ≥ √(2(k+1)n). This recovers the bound √(2kn) for “increasing‑first” k‑modal subsequences, a result that is slightly stronger (by a factor of √2) than the √((2k+1)n) bound originally conjectured.

To demonstrate that the √(2kn) bound cannot be substantially improved, the authors construct an explicit family of permutations via the Python routine strongMake(k, t). The construction consists of k blocks of t consecutive decreasing runs (and, when k is odd, an additional triangular block of decreasing runs of lengths t‑1, t‑2, …, 1). Within each block all elements are larger than those in later blocks, so any increasing‑first k‑modal subsequence can take at most one element from each decreasing run. A careful counting shows that the longest such subsequence has length at most k·t, while the total length of the permutation is about k·t²/2. Hence the longest increasing‑first k‑modal subsequence has length ≈ √(2kn), confirming the tightness of the bound.

The paper also proves a complementary statement: in any permutation there exists an index a_i that simultaneously serves as the endpoint of an increasing‑first k‑modal subsequence and a decreasing‑first k‑modal subsequence, each of length at least √(2kn) (up to constants). The proof again uses the labeling map, now defining two sets A(a,b) and B(a,b) that count points with a given x‑value or y‑value below a threshold. A combinatorial “trigger condition” is identified: if both |A(a,b)| > N+1‑a and |B(a,b)| > N+1‑b for some (a,b), then a longer subsequence can be built, contradicting maximality. By showing that any set of points avoiding this condition can contain at most N(N+1)/2 points, the same area argument yields the desired √(2n) lower bound for the simultaneous case. The induction on k proceeds analogously, giving the √(2kn) bound for the k‑modal version.

Methodologically, the paper’s contribution lies in replacing the traditional poset‑chain/antichain framework with a clean labeling‑to‑grid‑area argument. This approach not only reproduces known results but also yields a stronger bound for the “increasing‑first” variant and provides a unified treatment of the simultaneous increasing/decreasing endpoint problem. The authors suggest that such labeling techniques could be adapted to other combinatorial optimization problems on permutations, to probabilistic models of random permutations, and to algorithmic constructions where explicit bounds on modal subsequence length are required.

In summary, the paper delivers:

1. A new proof of Xu’s √((2k+1)n) lower bound for general k‑modal subsequences using labeling and lattice‑area estimates.
2. A stronger √(2kn) lower bound for increasing‑first k‑modal subsequences, together with an explicit construction showing near‑optimality.
3. A proof that some element of any permutation simultaneously supports long increasing‑first and decreasing‑first k‑modal subsequences.
4. A methodological shift that may inspire further research on permutation patterns, extremal combinatorics, and related algorithmic applications.