Papers

Socolar and Taylor asked for a single, simply connected three-dimensional prototile that forces nonperiodicity by shape alone, admitting no weakly nonperiodic tiling; the Schmitt-Conway-Danzer biprism and the three-dimensional Socolar-Taylor tile admit screw motions or a periodic stacking direction. We exhibit a rational polyhedral $3$-ball $Q$, which we call Chair44 (R44): a seven-cube chair whose $24$ exposed unit panels carry tiny square-pyramid features, and prove, as a proof submission, that $Q$ admits tilings of $\mathbb{R}^3$ by congruent copies, reflections allowed, and that every such tiling has no translational period and a symmetry group of order at most $24$; every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The solid was designed to a reading of the aperiodic-monotile phenomenon reached with the Six Birds emergence calculus (Section 3.3), and the construction turns on a single finite test, checked by machine: the tile's own contact rule survives coarsening, so that the decoded parent tiling obeys the tile's rule and no other. The proof combines a written geometric argument, that the features force every tiling onto a registered lattice, with exhaustive finite enumerations; the companion census is replayed by two independent implementations, every finite gate is kernel-checked in Lean 4 (modulo a named compiler hook per native_decide theorem), and the written geometric lemmas and the logical assembly are Lean theorems as well, so that the theorem is kernel-checked modulo the named compiler hooks; the written proofs remain as exposition.

We prove Chvátal's conjecture, posed in 1972: every hereditary family of subsets of a finite set has a largest intersecting subfamily that is a star. More generally, we prove a sharp correlation inequality for increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$. Writing $g^*(x)=1-g(1-x)$, we show that $$ \sum_{\varnothing\ne S\subseteq[n]}\hat{g}(S)^2\max_{i\in S}\mathrm{Inf}_i[f]\le\frac{2\mathrm{Cov}(f,g)\mathrm{Cov}(f,g^*)}{\mathrm{Cov}(f,g)+\mathrm{Cov}(f,g^*)}. $$ When $g$ is antipodal, that is, $g=g^*$, this yields $\mathrm{Cov}(f,g)\ge\frac{1}{4}\min_{i\in[n]}\mathrm{Inf}_i[f]$, the correlation formulation of Chvátal's conjecture due to Friedgut, Kahn, Kalai and Keller.

A prime circle of order $2n$ is a circular ordering of $1,\ldots,2n$ such that the sum of every two adjacent terms is prime. We prove that a prime circle exists for every sufficiently large $n$. The proof is based on a perfect matching and robust expansion. We also study Hamilton cycles in graphs and digraphs defined by primitive sums and differences over finite fields. In particular, the primitive-sum graph on $\F_q$ is Hamiltonian for every prime power $q>18\,888\,871$, and for the graph on a full prime field $\F_p$, the bound improves to $p>61$.

Given a finite dimensional Lie algebra $L$ on a finite field $\mathbb{F}_{p^n}$ of prime power order $p^n$ (with $n$ positive integer and $p$ prime), we consider the number of modular pairs $(A,B)$ in the lattice of all subalgebras $\mathcal{L}(L)$ and introduce the notion of “subalgebra commutativity degree” of $L$. This represents the probability to find that two randomly chosen subalgebras $A$ and $B$ of $L$ are permutable. We investigate the subalgebra commutativity degree of $L$ in connection with recent techniques of algebraic combinatorics and number theory, providing upper and lower bounds which may influence the structure of $L$. A specific study for the subalgebra commutativity degree of Heisenberg algebras is executed.

We propose a notion of simplicity for Cayley permutations that is compatible with inflation. We prove that Cayley permutations admit a substitution decomposition analogous to that of permutations and use it to enumerate simple Cayley permutations, primitive simple Cayley permutations, and simple restricted growth functions. We also prove that every hereditary Cayley permutation class with finitely many simple members has a finite basis and an algebraic generating function.

Difference sets, external difference families and near-factorizations of groups are much-studied structures satisfying certain uniformity conditions on the differences or sums within or between elements of sets in groups. Research has centred on abelian groups, though there are classic results (such as Dillon's Dihedral Trick for difference sets and the work of Pêcher on near-factorizations) connecting abelian and non-abelian structures, which have recently regained attention. We present a new explicit framework enabling transfer of difference structures between groups using a semidirect product approach, establishing new tools and constructions, encompassing various previous results and addressing an open problem of Swartz et al. Motivated by the classic dihedral results, we focus on semidirect products $G \rtimes \mathbb{Z}_2$. Transfer from abelian to non-abelian groups, and between distinct non-abelian groups, are both possible, and our framework can handle $\lambda$-fold near-factorizations, difference sets and external difference structures with arbitrarily many sets. We obtain new near-factorizations and difference structures in a range of groups, and can transfer various examples not transferrable by previous approaches. We showcase our approach by establishing a new infinite family of three-set abelian circular external difference families, then producing from this the first infinite family of non-abelian circular external difference families.

A set $S$ of vertices of a graph $G$ is a connected mutual-visibility set if every two vertices of $S$ are joined by a shortest path whose internal vertices lie outside $S$, and the subgraph induced by $S$ is connected. We introduce the connected mutual-visibility number $\mu_c(G)$, defined as the maximum cardinality of such a set, and investigate its structural and algorithmic properties. We establish fundamental bounds, derive Nordhaus–Gaddum type inequalities, and characterise the graphs attaining the minimum and maximum possible values. For regular $(d,2,-\delta)$-graphs, we derive general bounds on $\mu_c(G)$ and determine its exact value for the two cubic graphs of defect $2$. We further show that $\mu_c(G)$ is determined locally by the block structure of $G$, namely, it is equal to the maximum of the corresponding values over the blocks of $G$. Finally, we present a polynomial-time algorithm for recognising connected mutual-visibility sets and prove that the associated decision problem is $\mathsf{NP}$-complete, even for connected bipartite graphs of diameter at most $4$.

In this paper we prove a free-join decomposition theorem on Gorenstein polytopes significantly improving upon results in prior joint work of the second author with Batyrev, Borger, Haase, Kretschmer, and Payne. In particular, as a strong extension of the Batyrev-Juny lattice pyramid theorem it implies that any $d$-dimensional Gorenstein polytope $P$ of index larger than $\frac{d+2}{2}$ is a free join of Gorenstein polytopes. Here, the index (also called codegree) is the dilation factor $r$ such that $rP$ is reflexive. As our main application, we prove that the stringy $E$-polynomial of a Gorenstein polytope $P$ vanishes if and only if $P$ is thin (i.e., its local $h^*$-polynomial vanishes), and it has the expected degree otherwise. The latter was conjectured by Batyrev and the second author. The proofs of the general results were found via ChatGPT 5.6 Sol, while a proof of the extremal case of the Batyrev-Juny conjecture is contained in the master thesis of the first author.

A family $F$ of subsets of $[n]$ is 3AP-intersecting if every two members have intersection containing a non-trivial three-term arithmetic progression. We prove that there is an absolute constant $c>0$ such that any such $F$ has size at most $(\tfrac12 - c)2^n$. This is the first non-trivial progress towards a conjecture of Simonovits and Sós that the maximum possible size is $2^{n-3}$. More generally, we show the same bound for $H$-intersecting families whenever $H$ is a $3$-graph on $[n]$ with bounded codegrees. A clique shows that this is sharp, in that the bounded codegree assumption cannot be removed.

A hypergraph is said to be integral if all of its adjacency eigenvalues are integers. Recently, Portugal and Del-Vecchio in [Appl. Math. Comput. 504: 129507 (2025)] studied the integral hypergraphs and gave a characterization of integral hypercycles in three particular cases: $3$-uniform, $4$-uniform and $5$-uniform hypercycles. In the same article, they conjectured that the $k$-uniform hypercycle on $n$ vertices $\Cnk$ is never integral for $k<n-1$, when $n>6$. In this article, we confirm this conjecture and prove that for $2\le k\le n-1$, $\Cnk$ is integral if and only if $k=n-1$ or $(n,k)\in\{(4,2),(6,2),(6,3),(6,4)\}$. The proof begins with computing the complete adjacency spectrum of $\Cnk$ and then uses Niven's theorem, a cyclotomic-unit lemma, an elementary property of Euler's totient function, and the Galois symmetry of cyclotomic fields to complete it. Our result gives a complete characterization of $k$-uniform integral hypercycles on $n$ vertices.

A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all proper edge-colorings of $G$. We prove that, for every fixed $0<\alpha<1$, every properly edge-colored $n$-vertex graph with minimum degree at least $\alpha n$ satisfies $\operatorname{rpc}(G,c)\leq(1+o(1))n/2$, where the coefficient $1/2$ is best possible. We also determine $\operatorname{rpc}(G)$ asymptotically for every complete multipartite graph. If $G=K_{n_1,\ldots,n_r}$ has order $n$ and largest and smallest part sizes $M$ and $s$, respectively, then, uniformly over all choices of the number and sizes of the parts, $\operatorname{rpc}(G)=(1+o(1))\max\{\min\{\lfloor n/2\rfloor,n-M\},(n-s)/2\}$. The proof combines pseudorandom packings of globally rainbow linear forests with a decomposition into dense parts and prescribed avoidance for arbitrary dense graphs, and with reserved connectors and a direct dominant-part argument for complete multipartite graphs.

The tree-independence number $tree\text{-}\alpha(G)$, the induced matching treewidth $tree\text{-}\mu(G)$, and the sim-width $simw(G)$ are graph parameters defined in terms of tree or branch decompositions. We establish two polynomial bounds for the tree-independence number of induced $K_{t,t}$-free graphs, one in terms of sim-width and the other in terms of induced matching treewidth. Abrishami et al. (SIDMA, 2025) and Brettell et al. (EJC, 2025) asked whether bounded sim-width, together with the exclusion of an induced $K_{t,t}$, implies bounded tree-independence number. We answer this question by proving that, for integers $t\geq 2$ and $s\geq 1$, every induced $K_{t,t}$-free graph $G$ with $simw(G)\leq s$ satisfies $tree\text{-}\alpha(G)=O_t\left((s+1)^{2t^2-2t}\right)$. This also proves a polynomial strengthening of a conjecture of Bešter Štorgel et al. (arXiv, 2026) concerning induced $K_{1,t}$-free graphs and improves a theorem of Alon et al. (arXiv, 2025) by reducing the exponent from $3t^2+1$ to $2t^2-2t$. Alon et al. (arXiv, 2025) asked whether, for fixed induced matching treewidth, the tree-independence number is polynomially bounded in $t$. Using a VC-dimension argument, we answer this question affirmatively by showing that, for integers $\mu\geq 1$ and $t\geq 2$, every induced $K_{t,t}$-free graph $G$ with $tree\text{-}\mu(G)\leq\mu$ satisfies $tree\text{-}\alpha(G)=t^{O_\mu(1)}$.

The Hajnal-Szemerédi theorem guarantees an equitable $(r + 1)$-coloring of every graph of maximum degree at most $r$. We prove a version of this result for more general prescribed color class sizes, subject only to a natural upper bound.

Let $M=(E,\mathcal{I})$ be a matroid of rank $r$. Let $\mathcal{I}_k$ be the independent sets of size $k$, and let $I_k=|\mathcal{I}_k|$ and $I=|\mathcal{I}|$. We show that if every set $F\in \mathcal{I}_{r-1}$ is contained in at least $\delta$ bases, then $$\ln \left(\frac{I}{I_r}\right)\geqslant \frac{I_{r-1}}{I_r}\cdot \delta \ln \left(1+\frac{1}{\delta}\right).$$ In particular, we have $$\frac{I}{I_r}\geqslant 2^{I_{r-1}/I_r}.$$ By combining this result with several other ideas, we prove that if $G$ is a simple connected graph on $n$ vertices, and $F(G)$ and $T(G)$ denote its numbers of spanning forests and spanning trees, respectively, then $$\frac{F(G)}{T(G)}\geqslant \frac{F(K_n)}{T(K_n)},$$ where $K_n$ is the complete graph on $n$ vertices. Equality holds if and only if $G=K_n$.

A subset $X$ of a finite group $G$ is $k$-power-avoiding if for all $g\in G$ we have that $\{g,g^k\}\not\subseteq X$. The paper shows that if $G$ contains a $k$-power-avoiding subset $X$ with $|X|\geq |G|-c$, then the group $G^k$ generated by the $k$th powers of elements of $G$ has bounded order (in $k$ and $c$). We provide more detailed structural results when $c\leq 2$, and in particular we classify the groups which arise when $c\leq 2$ and $k$ is prime.

We explain how to derive the Robinson-Schensted-Knuth (RSK) correspondence, a fundamental bijection in algebraic combinatorics, from the Yang-Baxter equation. The Yang-Baxter equation arose in the study of quantum many-body systems and later became a cornerstone of the theory of solvable lattice models, particularly vertex models. In a vertex model, arrows occupy the edges of a grid, and each vertex carries a Boltzmann weight determined by the arrows on the four edges meeting at it. The weight of a configuration is the product of these local weights, and a partition function is the sum of the weights of all configurations with prescribed boundary conditions. For a vertex model whose partition functions are the Schur polynomials, the two sides of each instance of the Yang-Baxter equation admit exactly one weight-preserving matching of their summands. Carried across a grid, this forced matching is the classical RSK correspondence in the form of Fomin's growth diagrams. In natural coordinates the local matching becomes the combinatorial three-dimensional R, a set-theoretic solution of the Zamolodchikov tetrahedron equation. Its periodic closure returns the combinatorial R-matrices of one-row crystals. The forced matching is special to the Schur weights. For the Hall-Littlewood and q-Whittaker deformations and their spin versions, at generic parameter values no deterministic matching works for all boundary data. Reading each instance of the Yang-Baxter equation probabilistically, we replace the matching by a coupling of the two sides - a bijectivization, or probabilistic bijection - and obtain Markov operators that transport probability measures attached to vertex models. Iterated over the grid, these operators produce randomized RSK-type dynamics and interacting particle systems, including q-PushTASEP and the stochastic six-vertex model.

We give negative answers to Kolmogorov's rearrangement problem and Garsia's conjecture. We construct a complete uniformly bounded orthonormal system for which every rearrangement admits a square-summable series divergent almost everywhere. The finite construction uses two copies of the trigonometric system in different orderings. The main ingredient is a combinatorial lemma which guarantees a prescribed ordering along an arithmetic progression in at least one of two related permutations. Its proof uses Szemerédi's theorem and a counting argument. A Walsh variant gives $N$-term $\{\pm1\}$-valued systems with $L^2$ maximal norm at least $c\log\log N$ in every ordering, matching Bourgain's upper bound.

An $n$-barrycade of height $h$ is a set of $h$ permutations of $[n]$ such that all the prefix sums are different. This notion was described in 2020 by Richard Guy, together with the central problem to determine for which values of $n$ there exists a break-free $n$-barrycade, that is, one in which every possible prefix sum from $1$ to $\frac{n(n+1)}{2}-1$ is covered. The height of such barrycade would be $\frac{n+2}{2}$. We prove that barrycades of linear height exist for infinitely many sizes: there is a constant $c>0$ such that for infinitely many positive integers $n$ there exists an $n$-barrycade of height at least $cn$. We also explore an infinite variant of the problem and propose three constructions – greedy, grasshopper and precise grasshopper – that give increasingly more satisfying results. Along the way we encounter new integer sequences and formulate conjectures concerning omitted elements, missing partial sums, and word representations of infinite barrycades, which are firmly supported by computational data.

Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur's theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound is optimal. We disprove this conjecture and prove new bounds. In particular, we show that the maximum fraction of rainbow Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ lies between $9/22$ and $8/15$ asymptotically. Moreover, we study the problem in the general $k$-color setting and establish new non-trivial bounds.

A digraph is antidirected if every vertex has indegree zero or outdegree zero. Let $k\ge2$, and let $F$ be an antidirected forest with $k$ arcs and no isolated vertices. We prove that every digraph $D$ of order $n$ with more than $g_k(n):=2\max\left\{\binom{2k-1}{2}, (k-1)\left(n-\frac{k}{2}\right)\right\}$ arcs contains $F$ as a subdigraph. For $n\ge2k-1$, this threshold equals $2\mathrm{ex}(n,kK_2)$ and is attained by symmetric digraphs arising from extremal $kK_2$-free graphs. Consequently, the maximum directed extremal number over all such forests is $2\mathrm{ex}(n,kK_2)$. The proof combines a counting inequality for rooted antidirected forests, embeddings extending vertex-disjoint arcs, and vertex deletion. In the remaining case, the Gallai–Edmonds decomposition of the underlying graph gives the required bound on the number of arcs.

Recently, C. Li posted a preprint that resolves the fractional version of Talagrand's discrete convexity problem. In this note, we use the method of C. Li to answer another problem of Talagrand.

There is a natural connection between the SOS rank problem for bi-quadratic forms and the Zarankiewicz extremal problem for C4-free bipartite graphs. The classical Zarankiewicz number z(m,n) controls the bipartite skeleton associated with monomial squares. Allowing two cells to form a single bilinear square leads to augmented Zarankiewicz configurations and the second-order Zarankiewicz number z2(m,n). Unlike zRL and zSL, defined through recursive sufficient conditions, z2 maximizes over all irreducible displayed SOS decompositions without imposing (RW 3+). Hence, to prove z2(m,n)<=R, one must prove that every simple limited configuration with more displayed squares is reducible; failure of a sufficient condition cannot serve as a counterargument. We prove z2(4,4)=10, z2(7,4)=19, z2(8,4)=21, z2(5,5)=17, and obtain z2=zSL=zRL in all these cases. The extremal irreducible 6x4 configurations with 16 displayed squares form a single isomorphism class under row and column relabeling, whereas the extremal irreducible 7x4 configurations with 19 displayed squares form exactly three isomorphism classes. The four-column results form a structural chain: classify lower-order extremal configurations first, then use hereditary irreducibility under deletion of complete squares to constrain the next order. Finite exhaustive steps use candidate pruning, a necessary compatibility graph, clique enumeration, and orbit reduction, with a verifiable reducibility or irreducibility proof for each remaining orbit. For 5x5, there are two ordinary extremal skeletons; finite exclusion leaves only two highly symmetric 18-square candidates. They define the same ten-square polynomial, which admits an explicit nine-square representation, yielding the upper bound for z2(5,5).

The cocyclic development of Hadamard matrices has recently been extended from groups to loops by means of a cohomology theory that incorporates associativity obstructions. Over the finite field $\mathbb F_2$, the resulting loop-cocycles can be computed as solutions of homogeneous linear systems, making the framework suitable for exact computation. In this paper, we investigate multidimensional Hadamard matrices arising from finite loops. We introduce the notion of $\delta$-compatible pairs of $2$-cochains over a loop $L$ as a natural extension of the usual $2$-cocycle identity, which allows the use of two distinct $2$-cochains. They form a vector space naturally isomorphic to the direct product of the space of $2$-cocycles and the space of $1$-cochains. This yields a generalization of the classical cocyclic construction of multidimensional Hadamard matrices from groups to arbitrary loops, while preserving the computational advantages of the cocyclic approach. The obtained decomposition shows that the determination of Hadamard equivalence classes can be reduced from the full space of $\delta$-compatible pairs to the smaller space of ordinary $2$-cocycles, eliminating a redundant factor of $2^n$ from the search space. Computational results for all groups of orders $4$ and $8$, together with a non-associative loop of order $8$, show that the proposed three-dimensional construction refines the classical cocyclic Hadamard classification. While all examples collapse into a single equivalence class in dimension two, they split into several distinct classes in dimension three. In particular, the non-associative loop produces a three-dimensional Hadamard class that does not arise from any of the groups considered, showing that the multidimensional construction detects structural information that is invisible at the cocyclic matrix level.

We study the minimum adjacency spectral radius in the class $\mathcal{SC}_{m+2}(m)$ of strongly connected digraphs with $m$ vertices and $m+2$ edges. Using generating functions for directed paths, we associate with the relevant digraphs topological polynomials whose smallest positive roots determine the corresponding spectral radii. Based on an ear decomposition, we obtain a complete structural classification of $\mathcal{SC}_{m+2}(m)$ by showing that every digraph in this class can be obtained from a butterfly digraph by attaching a single ear. This reduces the extremal problem to the optimization and comparison of finitely many polynomial families subject to their realizability conditions. We prove that the minimum spectral radius is determined by the polynomial $P_{\min}(z)=1-2z^{m-1}-z^m$. If $R_m\in(0,1)$ denotes its unique root, then $\min_{G\in\mathcal{SC}_{m+2}(m)}\rho(G)=R_m^{-1}$. For $m\geq4$, the minimum is attained, up to isomorphism, uniquely by the cross-chorded cycle $\mathcal{C}_m^\times$. For $m=3$, there are exactly two non-isomorphic minimizers, both with spectral radius $(1+\sqrt5)/2$. Finally, we establish the bounds $2^{1/(m-1)}<\rho\left(\mathcal{C}_m^\times\right)<3^{1/(m-1)}$.

Backman and Liu proved that every integral generalized permutohedron of type $A$, and in particular every matroid base polytope, admits a regular unimodular triangulation. The analogous statement fails in type $B$: the delta-matroid simplex \[\operatorname*{conv}\{\mathbf{0},\ e_1+e_2,\ e_1+e_3,\ e_2+e_3\}\] has normalized volume $2$ and no lattice points other than its vertices, so it has no unimodular triangulation. We show moreover that, up to the natural symmetries of the $0/1$ cube and deletion of constant coordinates, it is the unique non-unimodular delta-matroid polytope that is a simplex. We prove instead that every delta-matroid polytope admits a regular dyadic triangulation, meaning a lattice triangulation whose maximal simplices have normalized volumes that are powers of two. More generally, every integral type $B$ generalized permutohedron admits such a triangulation. The main lattice-theoretic ingredient is that the type $B$ root configuration forms a totally dyadic system, a $2$-local analogue of total unimodularity. As a consequence, these polytopes satisfy a dyadic version of the integer decomposition property. In each dimension the corresponding exponent can be chosen uniformly, even though ordinary integer decomposition can fail for delta-matroid polytopes.

Partizan Edge Geography is a two-player game on a graph where each player has their own token on a vertex and moves their token to a neighbor in a turn removing the edge. Two player alternately move their tokens and the first player who cannot move loses the game. Fraenkel and Simonson (TCS, 1993) showed that the winner determination of this game is PSPACE-complete on directed graphs, given a graph and token positions. This paper resolves its complexity on undirected graphs by showing the PSPACE-completeness on bipartite undirected graphs of maximum degree 3. The same reduction also works for a variant where two tokens cannot be placed on the same vertex.

In this paper, we establish two Turán-type results for signed graphs. We first generalize the classical Turán theorem to signed graphs and then extend Nikiforov's spectral Turán theorem to signed graphs. Moreover, we determine the second maximum spectral radius among all unbalanced signed graphs that contain no balanced complete signed subgraph on \(r+1\) vertices.

We consider $n\times n$ covariance matrices $M=\frac{1}{n}XX^*$ where $X=(x_{i,j})$ is a matrix whose entries are independent complex random variables with $\mathbb{E}(x_{i,j})=0$ and $\mathbb{E}(|x_{i,j}|^2)=1$. We derive a $\frac{1}{n}$ expansion of the mixed moments, $\frac{1}{n}\mathbb{E}(\Tr(M^{(r_1)}\cdots M^{(r_q)}))$, of the form $a_0+a_1\frac{1}{n}+O(\frac{1}{n^2})$. This permits us to find explicit formulas for the moments and infinitesimal moments of several covariance matrices where we allow repetition. As an application of our formulas, we derive asymptotic freeness and infinitesimal freeness of independent covariance matrices under a fourth-moment condition. This generalizes previous results for the Wishart ensemble in which $x_{i,j}$ is complex Gaussian.

Let $f(k)$ denote the smallest integer such that every oriented graph $D$ with chromatic number at least $f(k)$ contains every oriented tree on $k$ vertices. Burr (1980) showed that $f(k)\le (k-1)^2$ and conjectured that $f(k)=2k-2$. Bessy, Gonçalves and Reinald (2025) proved that $f(k)=O(k^{3/2})$. In this paper, by using an absorbing set method, we show that $f(k)\le \lfloor 31\log (k!)\rfloor=O(k\log k)$.

In this paper, we build a Demazure crystal structure on the set of all flagged set-valued reverse plane partitions of a given skew shape $\lambda/\mu$ and flag $\Phi$. Our construction provides a unified generalization of the Demazure crystal structures previously known for the set of all flagged semi-standard set-valued tableaux and the set of all flagged reverse plane partitions. Consequently, we obtain a combinatorial expansion for the flagged hybrid Grothendieck polynomial $H_{\lambda/\mu}(\mathbf{x}_{\Phi};\mathbf{t};\mathbf{w})$, introduced by Guo–Kang–Liu, in terms of key polynomials. Applying this expansion, we express the hybrid Grothendieck polynomial $H_{\lambda/\mu}(\mathbf{x};\mathbf{t};\mathbf{w})$ in terms of both the stable Grothendieck polynomials $G_{\nu}(\mathbf{x})$ and the dual stable Grothendieck polynomials $g_{\nu}(\mathbf{x})$.