We investigate positivity and probabilistic properties arising from the Young–Fibonacci lattice $\mathbb {YF}$, a 1-differential poset on words composed of 1’s and 2’s (Fibonacci words) and graded by the sum of the digits. Building on Okada’s theory of clone Schur functions, we introduce clone coherent measures on $\mathbb {YF}$ which give rise to random Fibonacci words of increasing length. Unlike coherent systems associated to classical Schur functions on the Young lattice of integer partitions, clone coherent measures are generally not extremal on $\mathbb {YF}$. Our first main result is a complete characterization of Fibonacci positive specializations – parameter sequences which yield positive clone Schur functions on $\mathbb {YF}$. Second, we establish a broad array of correspondences that connect Fibonacci positivity with: (i) the total positivity of tridiagonal matrices; (ii) Stieltjes moment sequences; (iii) the combinatorics of set partitions; and (iv) families of univariate orthogonal polynomials from the (q-)Askey scheme. We further link the moment sequences of broad classes of orthogonal polynomials to combinatorial structures on Fibonacci words, a connection that may be of independent interest. Our third family of results concerns the asymptotic behavior of random Fibonacci words derived from various Fibonacci positive specializations. We analyze several limiting regimes for specific examples, revealing stick-breaking-like processes (connected to GEM distributions), dependent stick-breaking processes of a new type, or limits supported on the discrete component of the Martin boundary of the Young–Fibonacci lattice. Our stick-breaking-like scaling limits significantly extend the result of Gnedin–Kerov on asymptotics of the Plancherel measure on $\mathbb {YF}$. We also establish Cauchy-like identities for clone Schur functions whose right-hand side is presented as a quadridiagonal determinant rather than a product, as in the case of classical Schur functions. We construct and analyze models of random permutations and involutions based on Fibonacci positive specializations along with a version of the Robinson–Schensted correspondence for $\mathbb {YF}$.

We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogs of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the author. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.

A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov’s result in the following form. Given $r,t\in \mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include:

• • For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline {G}$ has positive triangle density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline {G}$ has positive $K_r$-density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2.

• • Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $\Omega ({n \over {\log n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $\Omega (n)$, where the constant $(2h-2)^{r}+1$ is optimal.

The ${n \over {\log n}}$ size of the blowups in all our results are optimal up to a constant factor.

We show that for any set $A\subset {\mathbb N}$ with positive upper density and any $\ell ,m \in {\mathbb N}$, there exist an infinite set $B\subset {\mathbb N}$ and some $t\in {\mathbb N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1<b_2 \}+t \subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1 \leq b_2 \}$, for infinite $B\subset {\mathbb N}$ and prove that any set $A\subset {\mathbb N}$ with lower density $\underline {\!\mathrm {d}}(A)>1/2$ contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed.

In this paper we study degree-penalized contact processes on Galton-Watson (GW) trees and the configuration model. The model we consider is a modification of the usual contact process on a graph. In particular, each vertex can be either infected or healthy. When infected, each vertex heals at rate one. Also, when infected, a vertex u with degree $d_u$ infects its neighboring vertex v with degree $d_v$ with rate $\lambda / f(d_u, d_v)$ for some positive function f. In the case $f(d_u, d_v)=\max (d_u, d_v)^\mu $ for some $\mu \ge 0$, the infection is slowed down to and from high-degree vertices. This is in line with arguments used in social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts.

We show that new phase transitions occur in terms of the parameter $\mu $ (at $1/2$) and the degree distribution D of the GW tree.

• • When $\mu \ge 1$, the process goes extinct for all distributions D for all sufficiently small $\lambda>0$;

• • When $\mu \in [1/2, 1)$, and the tail of D weakly follows a power law with tail-exponent less than $1-\mu $, the process survives globally but not locally for all $\lambda $ small enough;

• • When $\mu \in [1/2, 1)$, and $\mathbb {E}[D^{1-\mu }]<\infty $, the process goes extinct almost surely, for all $\lambda $ small enough;

• • When $\mu <1/2$, and D is heavier than stretched exponential with stretch-exponent $1-2\mu $, the process survives (locally) with positive probability for all $\lambda>0$.

We also study the product case, where $f(d_u,d_v)=(d_u d_v)^\mu $. In that case, the situation for $\mu < 1/2$ is the same as the one described above, but $\mu \ge 1/2$ always leads to a subcritical contact process for small enough $\lambda>0$ on all graphs. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.

We generalize the seminal polynomial partitioning theorems of Guth and Katz [33, 28] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb {R}^n$ of k-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma $ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec {q}$ of length r, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb {R}[X_1, \ldots , X_n]$ of degree at most D such that each connected component of $\mathbb {R}^n \setminus Z(P)$ intersects at most $\sim \frac {|\Gamma |}{D^{n - k - r}}$ elements of $\Gamma $. Also, under some mild conditions on $\vec {q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most D defined with respect to $\vec {q}$, such that each connected component of $\mathbb {R}^n \setminus Z(P')$ intersects at most $\sim \frac {|\Gamma |}{D^{n-k}}$ elements of $\Gamma $. To do so, given a k-dimensional semi-Pfaffian set $\mathcal {X} \subseteq \mathbb {R}^n$, and a polynomial $P \in \mathbb {R}[X_1, \ldots , X_n]$ of degree at most D, we establish a uniform bound on the number of connected components of $\mathbb {R}^n \setminus Z(P)$ that $\mathcal {X}$ intersects; that is, we prove that the number of connected components of $(\mathbb {R}^n \setminus Z(P)) \cap \mathcal {X}$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemerédi-Trotter-type theorems, and also prove bounds on the number of joints between Pfaffian curves.

Andrews and El Bachraoui [‘On two-color partitions with odd smallest part’, Preprint (2024), arXiv:2410. 14190] recently investigated identities involving two-colour partitions, with particular emphasis on their connection to overpartitions, and posed questions regarding possible companion results. Subsequently, Chen and Zou [‘Combinatorial proofs for two-colour partitions’, Bull. Aust. Math. Soc. 113(1) (2025), to appear] obtained some companion results by employing q-series identities and generating functions. In addition, they presented a combinatorial proof for one of their own results and one of the results of Andrews and El Bachraoui. They posed questions regarding combinatorial proofs of the remaining companion results. In this paper, we provide such proofs.

The study of Koszul binomial edge ideals was initiated by V. Ene, J. Herzog, and T. Hibi in 2014, who found necessary conditions for Koszulness. The binomial edge ideal $J_G$ associated to a finite simple graph G is always generated by quadrics. It has a quadratic Gröbner basis if and only if the graph G is closed. However, there are many known nonclosed graphs G where $J_G$ is Koszul. We characterize the Koszul binomial edge ideals by a simple combinatorial property of the graph G.

This paper is concerned with a duality between $r$-regular permutations and $r$-cycle permutations, and a monotone property due to Bóna-McLennan-White on the probability $p_r(n)$ for a random permutation of $\{1,2,\ldots, n\}$ to have an $r$th root, where $r$ is a prime. For $r=2$, the duality relates permutations with odd cycles to permutations with even cycles. For the general case where $r\geq 2$, we define an $r$-enriched permutation as a permutation with $r$-singular cycles coloured by one of the colours $1, 2, \ldots, r-1 $. In this setup, we discover a bijection between $r$-regular permutations and enriched $r$-cycle permutations, which in turn yields a stronger version of an inequality of Bóna-McLennan-White. This leads to a fully combinatorial understanding of the monotone property, thereby answering their question. When $r$ is a prime power $q^l$, we further show that $p_r(n)$ is monotone. In the case that $n+1 \not\equiv 0 \pmod q$, the equality $p_r(n)=p_r(n+1)$ has been established by Chernoff.

The purpose of this work is to develop a version of Forman’s discrete Morse theory for simplicial complexes, based on internal strong collapses. Classical discrete Morse theory can be viewed as a generalization of Whitehead’s collapses, where each Morse function on a simplicial complex $K$ defines a sequence of elementary internal collapses. This reduction guarantees the existence of a CW-complex that is homotopy equivalent to $K$, with cells corresponding to the critical simplices of the Morse function. However, this approach lacks an explicit combinatorial description of the attaching maps, which limits the reconstruction of the homotopy type of $K$. By restricting discrete Morse functions to those induced by total orders on the vertices, we develop a strong discrete Morse theory, generalizing the strong collapses introduced by Barmak and Minian. We show that, in this setting, the resulting reduced CW-complex is regular, enabling us to recover its homotopy type combinatorially. We also provide an algorithm to compute this reduction and apply it to obtain efficient structures for complexes in the library of triangulations by Benedetti and Lutz.

In his seminal work on partitions and divisor functions, MacMahon introduced the following two q-series:

\begin{align*}A_k(q):= \sum_{0 \lt s_1 \lt s_2 \lt \cdots \lt s_k} \frac{q^{s_1+s_2+\cdots+s_k}}{(1-q^{s_1})^2(1-q^{s_2})^2\cdots(1-q^{s_k})^2},\qquad\quad\quad\quad\\[6pt]C_k(q):= \sum_{0 \lt s_1 \lt s_2 \lt \cdots \lt s_k}\frac{q^{2(s_1+s_2+\cdots+s_k)-k}}{(1-q^{2s_1-1})^2(1-q^{2s_2-1})^2\cdots(1-q^{2s_k-1})^2}.\end{align*}

In 2013, Andrews and Rose proved that $A_k(q)$ and $C_k(q)$ are quasimodular forms of weight $\leq 2k$. Recently, Ono and Singh proved two interesting identities involving $A_k(q)$ and $C_k(q)$ and showed that the generating functions for the three-coloured partition function $p_3(n)$ and the overpartition function $\overline{p}(n)$ have infinitely many closed formulas in terms of MacMahon’s quasimodular forms $A_k(q)$ and $C_k(q)$. In this paper, we introduce the finite forms $A_{k,n}(q)$ and $C_{k,n}(q)$ of MacMahon’s q-series $A_k(q)$ and $C_k(q)$ and prove two identities which generalize Ono–Singh’s identities. We also prove some new identities involving $A_{k,n}(q)$, $C_{k,n}(q)$ and certain infinite products based on two Bailey pairs. Those identities are analogous to Ono–Singh’s identities.

In 2004, Herzog, Hibi, and Zheng proved that a quadratic monomial ideal has a linear resolution if and only if all its powers have a linear resolution. We study a generalization of this result for square-free monomial ideals arising from facet ideals of a simplicial tree. We give a complete characterization of simplicial trees for which all powers of their facet ideal have a linear resolution. We compute the regularity of t-path ideals of rooted trees. In addition, we study the regularity of powers of t-path ideals of rooted trees. We pose a regularity upper bound conjecture for facet ideals of simplicial trees, which is as follows: if $\Delta $ is a d-dimensional simplicial tree connected in codimension one, then reg$(I(\Delta )^s) \leq (d+1)(s-1)~+$ reg$(I(\Delta ))$ for all $s \geq 1$. We prove this conjecture for some special classes of simplicial trees.

We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers–Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behaviour of thecoefficients at $q=1$ is Gaussian.

We study the bilateral preference graphs $\mathit{LK}(n, k)$ of La and Kabkab, obtained as follows. Put independent and uniform [0, 1] weights on the edges of the complete graph $K_n$. Then, each edge (i, j) is included in $\mathit{LK}(n,k)$ if it is bilaterally preferred, in the sense that it is among the k edges of lowest weight incident to vertex i, and among the k edges of lowest weight incident to vertex j. We show that $k = \log(n)$ is the connectivity threshold, solving a conjecture of La and Kabkab, and obtaining finer results about the window. We also investigate the asymptotic behavior of the average degree of vertices in $\mathit{LK}(n, k)$ as $n\rightarrow\infty$.

In the 1980s, Erdős and Sós initiated the study of Turán problems with a uniformity condition on the distribution of edges: the uniform Turán density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least $d$ contains $H$. In particular, they asked to determine the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349 – 366] and [J. Eur. Math. Soc. 20 (2018), 1139 – 1159], while the latter still remains open. Till today, there are known constructions of $3$-uniform hypergraphs with uniform Turán density equal to $0$, $1/27$, $4/27$, and $1/4$ only. We extend this list by a fifth value: we prove an easy to verify sufficient condition for the uniform Turán density to be equal to $8/27$ and identify hypergraphs satisfying this condition.

In this paper, we study the existence of $k$-$11$-representations of graphs. Inspired by work on permutation patterns, these representations are ways of representing graphs by words where adjacencies between vertices are captured by patterns in the corresponding letters. Our main result is that all graphs are $1$-$11$-representable, answering a question originally raised by Cheon et al. in 2018 and repeated in several follow-up papers – including a very recent paper, where it was shown that all graphs on at most $8$ vertices are $1$-$11$-representable. Moreover, we prove that all graphs are permutationally $1$-$11$-representable – that is representable as the concatenation of permutations of the vertices – answering the existence question in extremely strong fashion. Our construction leads to nearly optimal bounds on the length of the words, as well. It can, moreover, be adapted to represent all acyclic orientations of graphs; this generalizes the fact that word-representations capture semi-transitive orientations of graphs. Our construction also adapts easily to other $k \geq 2$ as well, giving representations using a linear number of permutations when the best known previous bounds used a quadratic number. Finally, we also consider the (non-)existence of ‘even–odd’-representations of graphs. This answers a question raised by Wanless after a conference talk in 2018.

Fix integers $r \ge 2$ and $1\le s_1\le \cdots \le s_{r-1}\le t$ and set $s=\prod _{i=1}^{r-1}s_i$. Let $K=K(s_1, \ldots , s_{r-1}, t)$ denote the complete $r$-partite $r$-uniform hypergraph with parts of size $s_1, \ldots , s_{r-1}, t$. We prove that the Zarankiewicz number $z(n, K)= n^{r-1/s-o(1)}$ provided $t\gt 3^{s+o(s)}$. Previously this was known only for $t \gt ((r-1)(s-1))!$ due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of $s_i$, for example, it gives $z(n, K(2,2,7))=n^{11/4-o(1)}$ where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.

In this article we study the theories of the infinite-branching tree and the r-regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be realized by infinite ultraproducts of polynomial exact classes of graphs, and provide a characterization of the Morley rank of definable sets in terms of the degrees of polynomials measuring their non-standard cardinalities. This answers negatively some questions from [2], where it is asked whether every stable generalised measurable structure is one-based.

The famous Sidorenko’s conjecture asserts that for every bipartite graph $H$, the number of homomorphisms from $H$ to a graph $G$ with given edge density is minimised when $G$ is pseudorandom. We prove that for any graph $H$, a graph obtained from replacing edges of $H$ by generalised theta graphs consisting of even paths satisfies Sidorenko’s conjecture, provided a certain divisibility condition on the number of paths. To achieve this, we prove unconditionally that bipartite graphs obtained from replacing each edge of a complete graph with a generalised theta graph satisfy Sidorenko’s conjecture, which extends a result of Conlon, Kim, Lee and Lee [J. Lond. Math. Soc., 2018].

We give a presentation of the torus-equivariant (small) quantum K-ring of flag manifolds of type C as an explicit quotient of a Laurent polynomial ring; our presentation can be thought of as a quantization of the classical Borel presentation of the ordinary K-ring of flag manifolds. Also, we give an explicit Laurent polynomial representative for each special Schubert class in our Borel-type presentation of the quantum K-ring.