Papers

With the help of a summation formula for $q$-series and the creative microscoping method, we shall establish two parametric $q$-supercongruences. They are both modulo the third power of a cyclotomic polynomial. When $q\to1$, one of them is able to engender the following conclusion: for any prime $p\equiv2\pmod{3}$ and any nonnegative integer $s$ subject to $ s\leq (p-2)/3$, \[\sum_{k=s}^{(p+1)/3+s}(6k-1)\frac{(-\frac{1}{3})_{k-s}(-\frac{1}{3})_{k+s}(-\frac{1}{3})_{k}}{(k-s)!(k+s)!k!} \equiv 0\pmod{p^3}.\]

We study the Theta operators of D'Adderio-Iraci-Vanden Wyngaerd at $t=1$ and show that the power-sum-indexed operators $\Theta_{\mathsf p_k}|_{t=1}$ agree with a commuting family of derivations when restricted to symmetric functions of positive degree. Writing $\widetilde{h}_a$ for the modified Macdonald function indexed by the single row $(a)$ we establish that $\Theta_{\mathsf p_\mu}\widetilde{h}_a|_{t=1}$ is, up to a normalization, the single-row Macdonald cumulant of Dolęga. We use these results to show that $\Theta_{\mathsf p_\mu}\widetilde{h}_a|_{t=1}$ and $\Theta_{\mathsf e_\lambda}\widetilde{h}_a|_{t=1}$ are both sums of vertical-strip LLT polynomials indexed by certain plane trees. The former further shows that single-row Macdonald cumulants are LLT-positive, thereby yielding a stronger form of the higher-order Macdonald positivity conjecture of Dolęga when all shapes are single rows.

We prove that, whenever $ p \ge n^{-1/2 + o(1)} $, with high probability $ G(n, p) $ admits a fractional triangle decomposition, that is, a non-negative weight function on its triangles for which the total weight of all triangles containing each edge is equal to 1. This bound on $ p $ is optimal up to the asymptotic error term, improving upon the recent state of the art, due to Mahabaduge and Simkin, that $ p \ge n^{-4/11 + o(1)} $ suffices. Our main tool is a deterministic theorem guaranteeing the existence of fractional clique decompositions in all hypergraphs satisfying suitable `clique-regularity' properties. We prove this by analysing an extension (and generalisation to hypergraphs) of an algorithm proposed by Mahabaduge and Simkin, in which, at each time step, the discrepancy at each edge is spread among its containing triangles. By showing the concentration of the relevant quantities in random $ k $-uniform hypergraphs, we obtain for all $ k \ge 2 $ and $ r \ge k + 1 $ that w.h.p. $ G^{(k)}(n, p) $ admits a fractional $ K^{(k)}_r $-decomposition whenever $ p \ge n^{-\frac{r - k}{\binom{r}{k} - 1} + o(1)} $, which improves upon results of Delcourt, Kelly, and Postle, and is best possible up to subpolynomial factors.

Billera, Jia, and Reiner (2009) introduced the quasisymmetric functions of matroids and showed that this defines a Hopf algebra homomorphism which is a valuative invariant, i.e., isomorphic matroids have the same quasisymmetric function and polytopal subdivisions of matroid base polytopes define relations among the corresponding quasisymmetric functions. In this project we study an analogue in non-commuting variables, the word-quasisymmetric functions. To every matroid $M$ we associate a word-quasisymmetric function $\psi(M)$ and call this the chromatic word-quasisymmetric functions of a matroid. Matroids and word-quasisymmetric functions form Hopf algebras, and our map $\psi$ between them is a homomorphism. We want to study the kernel, equivalently the image, of the map $\psi$ from matroids to word-quasisymmetric functions, that is, we would like to understand which matroids are indistinguishable by the chromatic word-quasisymmetric functions. The map $\psi$ is not an invariant, but we can show that it is valuative. Using Schubert matroids and nested matroids, special classes of matroids, we prove a lower bound of $2^d-d$ for the rank of the map $\psi$ from matroids to the chromatic word-quasisymmetric functions in degree $d$ and conjecture the upper bound of $d!$ is tight.

In this paper, we establish a general identity for three-term recurrence sequences and then give applications to orthogonal polynomials and Apéry-like numbers.

For a $3$-graph $F$, the Turán number of $F$, denoted by $\ex(n,F)$, is the maximum number of edges in a $3$-graph on $n$ vertices containing no subgraph isomorphic to $F$. Let $F^-_{4,3}$ be the $3$-graph formed by a complete four-vertex core and three outer vertices, with all but one of the twelve triples containing one core vertex and two outer vertices. We prove that, for every $n\ge8$, \[ \ex(n,F^-_{4,3})=\binom n3-\binom{\lfloor n/2\rfloor}{3}-\binom{\lceil n/2\rceil}{3}, \] and the balanced complete bipartite $3$-graph is the unique extremal configuration. This determines the exact value and all equality cases in the asymptotic theorem of Mubayi and Rödl. It also extends the exact Turán Number of $F_{3,3}$ and resolves a conjecture of Frankl, Huang and Rödl.

Let $G= C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $\rr(G)=r$ denote its rank. The Davenport constant $\DD(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\DD^*(G)=1+\sum_{i=1}^r(n_i-1)$ is its classical lower bound. A long-standing conjecture asserts that $\DD(G)\le\DD^*(G)+\rr(G)-1$. In this paper, we disprove this conjecture. More strongly, we prove that $\sup_{\rr(G)=r}\bigl(\DD(G)-\DD^*(G)\bigr)=\infty$ for every fixed $r\ge5$. Thus the classical lower bound does not approximate the Davenport constant within an additive error depending only on the rank. More precisely, our construction gives $\delta(C_m^{r-4}\oplus C_{3m}^3\oplus C_{6m})\ge m$ for all integers $r\ge5$ and $m\ge2$. Consequently, the least rank $r_0$ for unbounded Davenport excess $\DD(G)-\DD^*(G)$ satisfies $3\le r_0\le5$. Our result also disproves the Narkiewicz–Śliwa conjecture of 1982 \cite{NS82} on the Narkiewicz constant, arising in algebraic number theory from the quantitative study of algebraic integers with unique factorization. The same amplification of the Davenport excess yields counterexamples to Girard's conjecture \cite[Conjecture 1.2]{Girard08} on the cross numbers of long zero-sum-free sequences. As a further main result, we establish the uniform upper bound $$\DD(G)\le\frac{16}{5}\rr(G)\exp(G)$$ for every nontrivial finite abelian group $G$. The classical estimate of van Emde Boas and Kruyswijk \cite{vEBK69} gives $\DD(G)\le\exp(G)\left(1+\log\frac{|G|}{\exp(G)}\right) \le\exp(G)\bigl(1+(\rr(G)-1)\log\exp(G)\bigr)$. Our bound removes the logarithmic factor $\log\exp(G)$ from this classical estimate, and replaces it with the absolute constant $16/5$.

We relate the theory of $q$-rational and $q$-real numbers introduced by Morier-Genoud and Ovsienko to the classical theory of Kleinian groups and their Teichmüller spaces. This provides a geometric point of view on several recent results about realisations of $q$-rationals for particular values of $ q \in \mathbb{C} $. As an application we resolve a conjecture of Bapat, Becker, and Licata on the topology of the $q$-deformed Farey tessellation.

We present various ways of determining generalized weight polynomials of a matroid $M$, and we recall how one can find the generalized weight spectra of a linear code, given these polynomials, for the matroid determined by any generator matrix of the code. A main goal is to give coding theorists different ways to determine these polynomials. We describe how one can find the generalized weight polynomials of any matroid $M$, directly from its lattice of flats, and we also show how one can find them from the Poincare series (in this case polynomials) of the associated Orlik-Solomon algebras of matroids arising as contractions of the flats of $M$. This opens for using information about broken circuits of the matroids to determine generalized weight polynomials. We also recall the well-known connection between the Orlik-Solomon algebra of a matroid, and Whitney numbers obtained by order homology, and use it to show how one can describe weight polynomials in terms of Whitney numbers from order homology of the matroid and its contraction of flats. We recall briefly a well-known relation between the Orlik-Solomon algebra of a matroid $M$, which is representable over the complex numbers, and de Rham homology numbers obtained from complements of intersections of hyperplanes in a hyperplane arrangement corresponding to $M$. We also describe a way to find the generalized weight polynomials of a matroid in terms of polynomials defined in connection with its lattice of cyclic flats. In a simple running example we show how one can calculate the generalized weight polynomials in different ways, including both the methods presented in this paper, and selected methods developed in earlier papers. We also include a less simple example with the projective Reed-Muller code $PR_3(2,2).$

Let $T_k(E)$ denote the set of congruence classes of ordered $k$-tuples of pairwise distinct points of $E$. Let $p\equiv3\pmod4$ be prime. For $E\subset\mathbb{F}_p^2$ with $3\leq|E|\leq p^{3/4}$, we prove that $|T_3(E)|\gg|E|^{11/6}$; for $4\leq|E|\leq p^{3/4}$, we prove that $|T_4(E)|\gg|E|^3/\log|E|$. For every fixed $k\geq5$ and $k\leq|E|\leq p^{3(k-2)/(3k-4)}$, we prove that $|T_k(E)|\gg_k|E|^{k-1}$. The proofs proceed by bounding the moments of the overlap function of rigid motions. The main inputs are an exact identity involving the distance energy and an incidence bound obtained by viewing rigid motions as lines over $\mathbb{F}_p(i)$. The bound for $k=4$ is sharp up to a logarithmic factor, while the bounds for $k\geq5$ are sharp up to constants.

How much information can a single neuron remember? How many memories can neural networks retrieve without creating false memories? These questions are related to a basic question: how many Boolean threshold functions $f(x)=\operatorname{sgn}(a_0+\langle a,x\rangle)$, $x\in\{-1,1\}^n$, are there? In this paper, we show that the number $T_n$ of distinct Boolean threshold functions is \[ T_n=2\binom{2^n-1}{n}\bigl(1+O(n^{-99})\bigr). \] Equivalently, the capacity of a single threshold neuron is $n^2-\log_2(n!)+1+O(n^{-99})$ bits, improving the $O(n)$ error term in the result of Kahn–Komlós–Szemerédi to $O(n^{-99})$. To prove this, we show that, for $1\le r\le n-1$, and $v_1,\ldots,v_r$ are chosen at random from $\{-1,1\}^n$, \[ \mathbb P\!\left\{ \langle v_1,\ldots,v_r\rangle\cap\{-1,1\}^n =\{\pm v_1,\ldots,\pm v_r\} \right\} =1-O(n^{-99}). \] In the context of the Kanter–Sompolinsky Hamiltonian for memory retrieval, this identifies $r=n-1$ as a sharp threshold, at which, for almost every collection of $r$ memories, the only ground states are these memories and their negatives, confirming a weaker form of the Kalai–Linial–Odlyzko conjecture. It also settles a recent open problem posed by M. Anthony on the specification number of Boolean threshold functions. In addition, we show that, for every $1\le r\le n-1$, \[ \mathbb P\{v_1,\ldots,v_r\text{ are linearly dependent}\} =2\binom r2\,2^{-n}+O\!\left(2^{-n}e^{-cn}\right), \] confirming a conjecture of Kahn–Komlós–Szemerédi.

Let $f(d)$ denote the smallest integer such that every bridgeless graph of diameter $d$ admits a strong orientation with diameter at most $f(d)$. It is known that $f(2)=6$ and $f(3)=9$. For $d=4$, the classical bounds of Chvátal and Thomassen [JCTB, 1978] imply $12\le f(4)\le40$, and subsequent work reduced the upper bound to 21. Very recently, Lin, Wang and You further established the substantially stronger bound $f(4)\le18$. Pushing this bound below $18$ turns out to be considerably more difficult, since the remaining extremal configurations cannot be handled by existing techniques based on $R-S$ orientations and related local constructions. In this paper, we prove that $f(4)\le16$. Our approach is entirely different from previous ones. Instead of constructing a strong orientation directly, we develop a sequential orientation framework together with auxiliary distance functions and a potential-function analysis. This enables us to control directed distances globally while avoiding the intricate case analysis required by earlier methods. We believe that the framework introduced here may be useful for studying oriented diameter problems of larger diameter.

We consider the Cartan square $V^{2\lambda}$ of a minuscule representation $V^\lambda$ of a simply laced complex simple Lie algebra $\mathfrak g$. We construct for $V^{2\lambda}$ a family of bases, which we call quiver bases, each indexed by the set $\operatorname{RPP}_2(P_\lambda)$ of reverse plane partitions of height two on the minuscule poset $P_{\lambda}$ of $V^{\lambda}$. Let $Q$ be a quiver on the Dynkin diagram of $\mathfrak g$, and let $c_Q$ be the corresponding Coxeter element. The quiver basis $\mathcal B^Q$ is distinguished by the following property: Up to sign, the action of the Tits representative $\dot c_Q$ on $\mathcal B^Q$ lifts the action of $c_Q$, via piecewise-linear toggles, on $\operatorname{RPP}_2(P_\lambda)$. This proves uniformly that, for any minuscule poset $P$, piecewise-linear Coxeter-motion and rowmotion on $\operatorname{RPP}_2(P)$ exhibit the cyclic sieving phenomenon. In type~$A$, the quiver basis for the standard orientation recovers, up to rescaling, the canonical basis, whose compatibility with the long cycle was established by Rhoades. In other types, however, we show the canonical basis is not compatible with any Coxeter element.

Building on the work of Pach, Rubin, and Tardos, we prove that every set of $n\ge2$ points in the real plane with no three collinear contains at least $cn$ pairwise crossing segments with distinct endpoints, for an absolute constant $c>0$.

We prove the coarse Erdős-Pósa conjecture of Georgakopoulos and Papasoglu. Informally, any graph either contains many fat cycles that are pairwise far apart, or there is a small number of bounded radius balls that together hit all of them. To be more precise, if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then there is a set $X\subseteq V(G)$ of $\mathcal{O}(k\log k)$ vertices such that every $q$-fat model of $K_3$ in $G$ has distance $\mathcal{O}(q)$ from $X$. In another form more closely resembling Manning's theorem that characterises quasi-trees: if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then $G$ is $\mathcal{O}(q)$-quasi-isometric to a graph $H$ that contains a set $X\subseteq V(H)$ of $\mathcal{O}(k\log k)$ vertices such that $H-X$ is a forest. Bootstrapping this result, we further prove that every graph with no $q$-fat model of $k \cdot K_3$ is quasi-isometric to a graph with no $k \cdot K_3$ minor, where the quasi-isometry can be chosen to only have additive distortion. This result also holds for $k = \infty$. We also obtain an Erdős-Pósa theorem for long induced cycles that are far apart.

Arrow patterns, introduced by Berman and Tenner, provide a unified framework for studying permutation classes where both one-line and cycle structure constraints are present. In this paper, we continue the systematic study of arrow pattern avoidance initiated by Archer and Laudone. We establish several structural results, including a key lemma that translates arrow patterns into vincular patterns under certain conditions, and derive a series of arrow-Wilf equivalences arising from reversal, complementation, and insertion operations. We also resolve the two cases $(12;3\to 3)$ and $(21;3\to 3)$ left open by Archer and Laudone, and enumerate the arrow patterns of the form $(\nu; b\to c)$ of size $3$ with $\nu \in \{31, 23, 32\}$ and $b,c\in [3]$, providing explicit formulas connecting the results to Bell numbers, Bessel numbers, Catalan numbers, and derangement numbers. Together with earlier work of Archer and Laudone, this leaves only $(32;1\to 3)$ unresolved for $|\nu|\le 2$, which we pose as an open problem.

A forest in an edge-colored graph is rainbow if its edges have pairwise distinct colors. We prove that for every $\varepsilon>0$ and all sufficiently large integers $m$, every properly edge-colored simple graph with $km$ edges, where $1\leq k\leq 2m$ and every color class has size at most $m$, contains at least $(1-\varepsilon)m$ pairwise edge-disjoint rainbow forests, each with exactly $k$ edges. The range $k\leq 2m$ is best possible: for every $k>2m$ there are such graphs containing no $k$-edge forest. Thus the conjecture of Montgomery, Pokrovskiy, and Sudakov fails beyond this range, while our theorem establishes its predicted conclusion throughout the largest possible range of $k$. The number of forests is asymptotically optimal. The proof uses an orientation dichotomy, hypergraph matching, matroid intersection, and martingale concentration.

In this paper we introduce and explore the following new graph invariant: For a graph $G$ on $k$ vertices, $G \neq K_k$, let $st(n,G)$ denote the maximum number of edges in a graph of order $n$ which does not contain any subgraph on $k$ vertices strictly containing $G$. A basic relation to classical Turán numbers is developed via the following: For $G$ on $k$ vertices, let $D(G) = \{ H : |H| = |G|, H = G + e \}$. Using this notion we prove that $ex(n,G) \leq st(n,G) = ex(n, D(G) ) \leq \min \{ ex(n,H) : H \in D(G) \}$ holds for all $n \geq |G|$. The family $D(G)$ happened to be smoothly amenable to the use of classical extremal results, and in many cases allows us to get asymptotically sharp estimates as well as exact values of $st(n,G)$. From the many results proved here we state the following as an illustration. (1) If $\chi(G) \geq 3$ and $\chi(D(G)) = \chi(G)$, then $st(n,G) = (1+o(1))ex(n,K_{\chi(G)})$. (2) If $\chi(D(G)) = \chi(G) +1$, then $G$ is a complete $\chi(G)$-partite graph and $st(n,G) = ex(n,K_{\chi(G) +1})$ for $n$ sufficiently large. (3) For $k$ odd, $k\geq 5$, $st(n,C_k) = ex(n,C_k) = ex(n,K_3)$ for $n$ sufficiently large. (4) If $T$ is a tree of order $q$ with diameter $k \geq 2$ and $q \geq k+1 \geq 3$, then $ex(n, \{C_3,...,C_{k+1}\}) \leq st(n,T) \leq ex(n, \{C_3,...,C_{k+1}\}) + (q-1)n$. Many results concerning even cycles, theta graphs, dense bipartite graphs and graphs of the form $G = G^* \cup tK_1$ are obtained, moreover the value of $st(n,G)$ is computed for all graphs on at most 4 vertices.

We prove the Marcus-Minc conjecture: let n be any integer at least 2, and let A be a nonnegative n by n matrix whose row and column sums are all one. Form a new matrix by subtracting A from the n by n matrix of ones and dividing by n minus one. Then the permanent of A is at least the permanent of the new matrix. We also determine all equality cases.

For binary matroids \(M,N\) representable over a common field \(\F\), the Kronecker product of their \(\F\)-representations defines a matroid \(T_{\F}(M,N)\) independent of the chosen representations. We classify when this represented tensor product is regular, cographic, graphic, or binary. For simple nonfree factors, regularity holds exactly when, up to interchange, one factor is a cactus matroid and the other is outerplanar, or one is a triangular cactus matroid and the other is series–parallel. Cographicity holds exactly in the first case. For simple factors with nonempty ground sets, graphicity holds exactly when one factor is free and the other is graphic. For fixed factors, regularity, cographicity, and graphicity are independent of the common representation field, although the isomorphism type may vary with its characteristic. Products of at least three simple nonfree factors are nonregular.

Variable-weight geometric orthogonal codes (GOCs) were introduced by Doty and Winslow for designing the macrobonds in 3D DNA origami. Recently, Wang et al. introduced the concept of geometric difference families (GDFs) and established the equivalence between GOCs and a certain type of GDFs. Based on the relationship, Su et al. gave two classes of variable-weight GOCs. In this paper, we first prove the existence of (n*m,K,1)-GDFs for K ={3,5} and {3,6}. For further research, we completely determine the existence of all the remaining parameters for an (n*m,{3,4,5},1)-GDF by Su et al. As consequences, the perfect (n*m,K,1)-GOCs for K = {3,5}, {3,6} and {3,4,5} are obtained.

We present a new proof of a classical theorem of Dieudonné: if a linear space of $n\times n$ matrices consists entirely of singular matrices, then its dimension is at most $n^2-n$. Our proof is based on a surprising “duality” argument: we prove this universal upper bound by exhibiting a single matrix space that serves as a lower bound for a related problem. Interestingly, this approach only works for certain fields, but we use model-theoretic arguments to obtain the same result for all fields. We hope that this approach can be generalized to provide new duality-based proofs of other classical theorems on matrix spaces, and give some preliminary results in this direction.

In this paper, we investigate the existence of pairs of orthogonal quantum Latin squares, each with maximal cardinality. Using characters and full-rank factorizations of finite abelian groups, together with a product construction, we establish two infinite families of orders for which such pairs exist.

This paper studies the antichain polynomials of $[k]\times P$, where $P$ is a connected minuscule poset. We give a formula for the number of antichains, counted by size, of an arbitrary poset. Using this formula, we present necessary and sufficient conditions for the palindromicity of antichain polynomials for two infinite families of connected minuscule posets. We show that, for every connected minuscule poset $P$, if the antichain polynomial of $[k]\times P$ is palindromic, then it has only real and strictly negative zeros. This result, in particular, gives an affirmative answer to Ding-Dong's conjecture about $\gamma$-positivity of the antichain polynomial of $[k]\times P$. By constructing a bijection between antichains and labeled Dyck paths, we also give a layer-refined enumeration of the antichains in $[2]\times[m]\times[n]$. We establish a relation between such antichains and Clar covers of the hexagonal flakes $O(2,m,n)$, thus prove a conjectured determinantal formula for a family of Zhang-Zhang polynomials. Real-rootedness and stability results are also obtained when the shortest chain has length at most two. Finally, we present infinitely many connected Peck posets whose antichain polynomials are not unimodal, disproving the log-concavity conjecture of Ding and Dong.

Using refined absorption, we prove that for every integer $g\ge 1$ and real $\gamma > 0$, and for sufficiently large $n$, there exists an $n^{-1+\gamma}$-spread distribution on Latin squares of order $n$ and girth at least $g$ that have no proper subsquares. This implies a combinatorial proof of Hilton's conjecture from the 1970s (recently proved algebraically by Allsop and Wanless) that for all sufficiently large $n$, there exists a subsquare-free Latin square of order $n$; indeed, it implies there exist at least $n^{(1-o(1))n^2}$ subsquare-free squares. Simultaneously it also implies the existence of high girth Latin squares (recently proved by Kwan, Sah, Sawhney and Simkin) and even an $n^{-1+\gamma}$-spread distribution on high girth Latin squares.

We develop a Ramsey theory for leaf-generated subsets of finite products of trees. Our starting point is a Theorem of Furstenberg and Weiss which states that for every $k\geq 1$ and $\alpha>0$, if $A$ is a subset of the leaves of $T_n$, where $T_n$ is the complete binary tree of height $n$, with size $|A|\geq 2^{\alpha n}$, then for $n$ sufficiently large the ancestor closed sub-tree $T_A\subset T_n$ generated by $A$ must contain a copy of $T_k$ such that (1) all vertices in the same level of $T_k$ are mapped into vertices at the same level of $T_A$, (2) if a non-leaf vertex $x\in T_k$ is mapped into a vertex $y \in T_A$, then the two children of $x$ are mapped into descendants of the two children of $y$, and (3) the levels of $T_A$ occupied by the copy of $T_k$ form an arithmetic progression. For the product $T_n \times T_n$ of two binary trees, we show that for every $k\geq 1$ and $\alpha > 1$, for $n$ sufficiently large, every subset $A$ of the leaves of $T_n\times T_n$ of size at least $2^{\alpha n}$ has that its ancestor closure $\Gamma_A\subset T_n \times T_n$ contains similarly structured arithmetic copies of the height $k$ four-ary tree, where the four children of every non-leaf vertex in the source tree are required to map below the four distinct diagonal children of the image vertex, where a diagonal child of a vertex $(x,y) \in T_n \times T_n$ is obtained by moving one level down each in each component. Our results generalise to product of $d$-many finite $b$-ary trees with a sharp critical exponent of \[ \alpha_{\text{crit}}(d,b) =d-1 + \log_b(b-1).\] Geometrically, our results imply that any set $E\subset [0,1)^d$ of upper Minkowski dimension greater than $\alpha_{\text{crit}}(d,b)$ contains arithmetic $b$-adic branching patterns of arbitrary finite order.

We prove that every $(2K_2,K_4)$-free graph is recolorable. Equivalently, for every such graph $G$ and every $\ell\geq \chi(G)+1$, the reconfiguration graph of proper $\ell$-colorings of $G$, in which two colorings are adjacent if they differ on exactly one vertex, is connected. This resolves the final remaining open case in the classification of recolorable $(F_1,F_2)$-free graphs when $F_1$ and $F_2$ have at most four vertices.

Motivated by a word encoding of tree-child networks, we introduce trident tableaux, which are Young tableaux with a unique three-cell column satisfying certain conditions on successors. We construct a bijection between trident tableaux with $n+1$ columns and two-wall tableaux with $n$ columns, that is, two-row fillings with two designated columns in which vertical order is not imposed. The bijection matches the three-part decompositions of the two classes and shows that each class has cardinality $n(n+1)C_n/2$, where $C_n$ is the $n$-th Catalan number. In particular, it gives a trident-tableau interpretation of a shifted form of OEIS A002457. As a consequence, we obtain an exact enumeration of tree-child networks with one reticulation node that contain a trident, and show that their proportion among all tree-child networks with one reticulation node tends to $1/8$ as the number of leaves tends to infinity.

How many equilateral points can $\ell_p^n$ contain beyond $n+1$? For every fixed real $p>4$, we construct equilateral sets for which the number of points beyond $n+1$ grows at least linearly with $n$. For every fixed dimension, we prove that the maximum is $n+1$ at all sufficiently large finite $p$, with an explicit threshold. The least dimension admitting any fixed number of extra points is $p^{1+o(1)}$ as $p\to\infty$. The constructions and finite bounds are formalised in Lean. Thus $p=4$ marks the boundary: $n+1$ is exact in every dimension for $2\leq p\leq4$, while at every fixed finite $p>4$ the excess grows at least linearly with dimension.

For a group $G$ and subsets $S,T \subset G$ we consider a family of mirror di-Cayley graphs $MX(G;S,T)$ and mirror di-Cayley sums graphs $MX^+(G;S,T)$, namely those with $T=\{e\}, S$ or $S \cup \{e\}$. We refer to them indistinctly by $MX^*(G;S,T)$. We can think of $MX^*(G;S,T)$ as two copies of the Cayley (sum) graph $X^*(G,S)$ joined by edges determined by the connection set $T$. Recently, in the work Isospectral Cayley graphs with even and odd spectrum, we study the spectrum of these graphs and several isospectrality problems. Here, we compute the energy and spectral $k$-moments of $MX^*(G;S,T)$ in terms of those of the underlying Cayley graphs $X(G,S)$. Then, we study energetic problems like hypo-, order-, and hyper-energeticity of these graphs. Finally, we give conditions for the existence of equienergetic pairs of non-isomorphic MDCGs.