Papers

Let $b(n)$ denote the minimum number of perfect matchings in a simple brace of order $n$. De~Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 27(2): (2013) 940-958) proved that $b(n)\ge(n-2)^2/8$ and conjectured that $b(n)=(n-2)^2/4$ for all sufficiently large even $n$. We prove this conjecture.

After centering and scaling, the length of the longest weakly increasing subsequence of a finite-state Markov random word converges to a Brownian last-passage functional with dependent components. We investigate when this functional, and more generally the limiting RSK Young-diagram shape, admits a Gaussian random-matrix representation. We introduce Hermitian Gaussian matrices with correlated diagonal entries and show that isotropy of the Brownian covariance on the trace-zero hyperplane yields such a representation. Within this natural class of Hermitian models with independent Gaussian off-diagonal entries, we prove a rigidity theorem showing that isotropy, together with equality of the off-diagonal variances, characterizes the conditional largest-eigenvalue representation. Explicit three-letter non-cyclic Markov chains demonstrate that this representation is not universal: the limiting Brownian functional may fail to have the law of the largest eigenvalue of any centered jointly Gaussian $3\times 3$ Hermitian matrix. Finally, we extend the random-matrix identification to chains with non-uniform stationary distributions and to countable state spaces.

We study quantum-assisted balanced rendezvous on the $d$-dimensional hypercube in a one-step two-agent quantum-walk framework, where the two agents move the same Hamming distance and meet at a common midpoint. The key structure is a projective $t$-sphere resolution, which governs exact finite-dimensional quantum attainment. Its scalar analogue recovers the corresponding classical characterization. We characterize finite-dimensional quantum attainment of the no-signalling benchmark, construct quantum strategies from projective sphere resolutions, and establish a rigidity theorem for optimal finite-dimensional strategies. A marked difference appears between nearest-neighbor and higher-cardinality updates: quantum advantage can occur in the former, whereas in the latter the geometry of the relevant Hamming spheres rules out exact finite-dimensional attainment of the no-signalling value. Together, these results connect quantum-walk dynamics, Hamming-sphere geometry, and quantum rendezvous within the quantum Bernoulli noises (QBN) framework.

Permutons can be thought of as "continuum limits" of permutations of diverging size, which encode information about the number of occurrences of any given finite subpattern. We construct a natural composition operation on permutons, analogous to composition of permutations, and explore its properties via a correspondence between permutons, Markov kernels, and Markov operators. We show that no extension of permutation composition to permutons can be (jointly) continuous with respect to the usual "weak" topology on permutons, but we construct a "strong" topology with respect to which composition is jointly continuous. Moreover, we characterize strong convergence of sequences of permutations of diverging size: a sequence of permutations converges strongly if and only if it converges weakly and its limit is the transpose of the graph of a measure-preserving function. The key insight behind these results is to apply functional analysis to certain operators associated to permutons; with this shift in perspective, our results have elegant and relatively "soft" proofs. As an application, we describe the permuton limit of a product of finitely many independent uniformly sampled separable permutations and give a formula for the expectation of this limit. We also show that in this case, as the number of compositions goes to infinity, the aforementioned expectation tends to the uniform permuton.

Let $F_5$ denote the $3$-uniform hypergraph with edge set $\{abc,abd,cde\}$. For a hypergraph property $\mathcal P$, let $ex_{\mathcal P}(n,F_5)$ be the maximum number of edges in an $n$-vertex $F_5$-free $3$-uniform hypergraph satisfying $\mathcal P$. Let $\Delta_2(H)$ denote the maximum codegree of a pair of vertices in a $3$-uniform hypergraph $H$. We determine the exact value of $ex_{\Delta_2\ge m}(n,F_5)$ when $m$ is bounded away from $n/3$ by a fixed positive proportion. More precisely, for every constant $c\in (1/3,1)$, there exists $n_0$ such that for all $n\ge n_0$ and $m=cn$, \[ ex_{\Delta_2\ge m}(n,F_5) = m\left\lceil\frac{n-m}{2}\right\rceil \left\lfloor\frac{n-m}{2}\right\rfloor. \] The unique extremal hypergraph is the complete tripartite $3$-graph $K^3_{m,\lceil (n-m)/2\rceil,\lfloor (n-m)/2\rfloor}$. We also prove the corresponding exact result for the blow-ups $F_5^t$, together with stability theorems showing that every near-extremal configuration is close to the relevant tripartite construction.

For every $d\ge4$, every prescribed connectivity $r\ge d$, and every $\varepsilon>0$, we construct a simplicial $d$-polytope whose graph has connectivity exactly $r$, is nontraceable, and has toughness less than $\varepsilon$. Thus the classical theorems of Whitney (1931) and Tutte (1956), which guarantee hamiltonicity under sufficiently high connectivity in dimension three, have no higher-dimensional counterpart. This answers a question posed by Bojan Mohar in 2001 on whether sufficiently high connectivity guarantees hamiltonicity of $4$-polytopal graphs and the connectivity questions raised by Ikegami, Maezawa, and Zamfirescu in 2021. For fixed $d\ge4$, $r\ge d$, and integer $k\ge2$, the examples can be chosen to have no closed spanning walk with at most $k$ visits per vertex, no spanning tree with degrees at most $k$, and nontraceable prisms. For fixed $d$ and $r$, we obtain graphs on $n$ vertices whose longest paths and cycles contain $O(\sqrt n)$ vertices. Consequently, the family of simplicial $d$-polytopes with connectivity exactly $r$ has Grünbaum–Walther shortness coefficient zero and shortness exponent at most $1/2$.

Given an integer $k \geq 2$ and a prime $p$ large enough depending on $k$, we show that, if $A \subseteq \mathbb F^n_p$ does not contain a restricted polynomial progression of the form \begin{align*} x, x+\begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, x+\begin{pmatrix} y_1^2 \\ y_2^2 \\ \vdots \\ y_n^2 \end{pmatrix}, \ldots, x+\begin{pmatrix} y_1^{k-1} \\ y_2^{k-1} \\ \vdots \\ y_n^{k-1} \end{pmatrix}, \end{align*} where $x \in \mathbb F^n_p$ and $y \in \{0,2, 3, \ldots, M_k\}^n \backslash \{\vec{0} \}$, for some fixed $M_k$, then $A$ must have density bounded above by $\frac{C}{(\log^{(k^{O(k)})} n)^c}$, where $C, c>0$ only depend on $k$ and $p$. For $k\geq 4$, the best previously known bound came from the density Hales–Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.

Alon, Bishnoi, Das and Neri constructed sequences of strong blocking sets in $(k-1)$-dimension projective space $P({\bf F}_q, k-1)$ with sizes at most $Cqk$, where $k$ goes to the infinity and $C$ is a fixed positive constant not depending on $q$, in their paper published in Transactions AMS, vol. 277, pp. 5389-5410, 2024. This is equivalent to a sequence of minimal $[n_i,k_i]_q$ codes with positive rate $\frac{k_i}{n_i} \geq c$, where $k_i$ goes to the infinity and $c$ is a fixed positive constant only depending on $q$. They solved one of the main open problems on minimal codes. The Alon-Bishnoi-Das-Neri construction was based on asymptotically good sequences of algebraic geometric codes and expander graphs. However, the constant $C$ is large for small prime power $q$. In this paper, based on a simple approach using $(q-1)$-power intersecting algebraic geometric codes via concatenation to construct minimal codes, we construct explicit strong blocking sets in $(k-1)$-dimension projective space with sizes around $2(q^2-1)k$, equivalently, explicit minimal $[n,k]_q$ codes with length around $2(q^2-1)k$. This is an essential improvement on the previous result of Alon-Bishnoi-Das-Neri, when $q \leq 11$. For large $q$, strong blocking sets and minimal codes in our construction are worse than the previous construction of Alon-Bishnoi-Das-Neri. However, our approach is much simpler and does not depend on expander graphs.

Using the spectral relation between irreducible cyclic codes (ICC) and generalized Paley (GP) graphs recently found by the authors, we study the weight distribution of irreducible cyclic codes of length $2^\ell$, $\ell \in \mathbb{N}$. These codes were previously studied by Sharma, Bakshi and Raka, who divided them into two families (each having two cases). First we characterize the GP-graphs spectrally related to these families of ICCs: they are complete, semiprimitive, Hamming, or Cartesian products of semiprimitive GP-graphs. We then compute the spectrum of these graphs and hence, by the mentioned spectral relation we finally obtain neat explicit expressions for the weight enumerators of all the irreducible cyclic codes of length $2^\ell$ with $\ell \in \mathbb{N}$.

Let $\Lambda_n$ be the set of all eigenvalues of standardized Laplacian matrices of order $n$. In 2005, a polygon $S_n\subseteq\Lambda_n$ was constructed, and it was conjectured that $\Lambda_n=S_n$. We observe that $S_n$ is a zonotope, which gives a short proof of $S_n\subseteq\Lambda_n$, and we describe $\Lambda_n$ as a union, over all vectors $v$, of intersections of zonotopes $Z_i(v)$ built from the ratios $v_j/v_i$. For a vector whose components are the $n$ distinct $n$th roots of unity, every $Z_i(v)$ equals $S_n$. We then show that the conjecture is false for $n=7$: the directed $6$-cycle with one vertex duplicated has an eigenvalue lying $2.1\cdot10^{-4}$ beyond one edge of $S_7$. For the analogous digraph of order $n$, the corresponding eigenvalue lies at signed distance $\delta_n\sim\frac{\pi^2}{2n^4}$ beyond the supporting line of the analogous edge of $S_n$, so the construction yields counterexamples for all sufficiently large $n$; numerically, it does so for every $7\le n\le300$. We also prove that $S_m\subseteq S_n$ whenever $m\mid n$, with $S_m\setminus\{0,1\}$ lying in the interior of $S_n$ for $n\ge3m$.

A geometric $(n_4)$ configuration is a set of $n$ points and $n$ lines in the real projective plane such that every point lies on exactly four of the lines and every line contains exactly four of the points. The question of which $n$ admit such a configuration was posed by Grünbaum in 2000, and successive work of Bokowski, Grünbaum, Schewe, Pilaud and Cuntz settled every case but $n=23$. We settle this last case by bounding the symmetry of such configurations and classifying the most symmetric ones exactly. We prove that for every prime $n\equiv3\pmod4$ the group of projective symmetries of a geometric $(n_4)$ configuration has order at most four, explaining why symmetric constructions failed at $n=23$, and the same argument bounds the symmetry of $(n_5)$ configurations for primes $n\not\equiv1\pmod5$. Reducing the Klein four-group case to 4021 bilinear polynomial systems and solving all of them exactly with Gröbner bases, we show that exactly two $(23_4)$ configurations reach this bound up to projective equivalence and that there are no others even over $\mathbb{C}$. Both are self-polar; the first has integer coordinates and the second is defined over $\mathbb{Q}(\sqrt{17})$ but not over $\mathbb{Q}$. Consequently geometric $(n_4)$ configurations exist if and only if $n\ge18$ and $n\ne19$. The same classification recovers Cuntz's $(22_4)$ and $(26_4)$ configurations and finds three new combinatorial types at $n=22$ and $94$ at $n=26$. Among self-polar $(23_4)$ configurations with a single involution we find $50$ more, defined over number fields of degree at most $7$. Four of them deform in one-parameter families, and computing three of these exactly, one birational to the modular curve $X_1(11)$, we prove that there are infinitely many geometric $(23_4)$ configurations, among them infinitely many with rational coordinates that are self-polar and have no projective symmetry.

We study a model of a random centrally symmetric polytope and ask about the existence of points in the polytope with non-zero, integer coordinates. Our main results are the location of a sharp threshold for the existence of these integer points, a characterization of the typical empirical distribution of their coordinates, and showing that near the feasibility threshold these integer points are almost always totally isolated — at linear Hamming distance to the nearest other integer point in the polytope. The behavior of these random integer programs is related to that of the symmetric binary perceptron (the case when coordinates of points are $\pm 1$-valued). The lack of symmetry in the set of potential solutions leads to new questions and phenomena, such as the distribution of the coordinate sizes in typical solutions, and requires the development of new information-theoretic techniques to deal with higher-dimensional optimization problems arising in the method of moments.

Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), p(n,2), ..., p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for every n >= 1. For n >= 10^5 we give an analytic argument with fully explicit constants: an exact probabilistic representation of the ratio p(n,k+1)/p(n,k), a second-order saddle-point main term with a rigorous uniform error bound in the transition window, and separate elementary or saddle-point estimates in the two tails. Every numerical constant in this argument is certified in ball arithmetic. For n <= 2*10^5 the conjecture is verified exhaustively by two independent programs, one of them using exact integer arithmetic only. Since log-concavity of k -> p(n,k) fails for every n >= 4, the argument works with first differences and never uses log-concavity.

We propose constructions of spherical $5$- and $7$-designs in odd dimensions by lifting balanced halves of even-dimensional designs to several layers of suitable latitudes. For every odd dimension, we obtain an explicit spherical $5$-design with at most $60(d^2-1)$ points, attaining the optimal quadratic order in the dimension. Assuming Zauner's conjecture, this bound improves to $7.5(d-1)^2$. For every odd $d \geq 9$, we also construct a spherical $7$-design with $110M$ points from a complex projective $3$-design with $M$ points in $\mathbb{CP}^{(d-3)/2}$.

A stable matching in a bipartite preference system is a matching that has no unmatched pair whose endpoints both prefer each other to their current assignments. Rothblum gave a linear description of the stable marriage polytope, the convex hull of the incidence vectors of stable matchings. We prove that this polytope is box-totally dual integral. The proof uses the affine representation of the stable marriage polytope as an order polytope together with the matricial characterization of box-TDI polyhedra. Since Kiraly and Pap proved that Rothblum's description is totally dual integral, our result further implies that Rothblum's system is box-totally dual integral.

This paper studies an open network design problem of constructing graph topologies with maximum algebraic connectivity under fixed numbers of vertices and edges. A density-dependent approach is proposed that decomposes the design task into two stages: a high-quality, low-cost structural backbone construction and a reinforcement learning completion stage. The method selects the backbone based on density, using complete multipartite graphs in dense regimes and Cayley graphs in low-to-intermediate regimes, and then refines the remaining edges through a learning-based procedure. Numerical results show that the proposed method achieves high algebraic connectivity with favorable runtime scalability, outperforming existing methods. These results support a structural-family-first strategy for scalable network design.

We prove that every finite ordered set of width at most three and cardinality at least four is reconstructible from its deck. This settles the width-three part of an open problem posed by Schröder in Ordered Sets; the separate question concerning ordered sets of height one is not addressed here. The new ingredient is a criterion for recovering the marked maximal deck of an ordered set with exactly three maximal elements from its unmarked maximal deck and the isomorphism type of the suborder obtained by deleting all maximal elements. We express the criterion as an orbital balance condition on the attachment ideals in this common base. Necessity follows by double counting; sufficiency uses the structure of automorphism actions in width at most three. The full-deck theorem then follows from reconstruction results of Schröder concerning maximal cards and marked maximal cards. A local reattachment argument correcting one intermediate step in the two-maximal-element case is included in an appendix.

In characteristic zero, we prove that the Tor algebras of the generic, symmetric, and skew-symmetric determinantal rings are Koszul with respect to a natural regrading. In particular, their Eilenberg–Moore spectral sequence degenerates, which allows us to explicitly determine the Poincaré series of their residue fields and answers a problem raised by Avramov. To do this, we construct the universal hook algebra and prove that it has a quadratic Gröbner basis and thus is Koszul. This leads to a new standard monomial construction of the irreducible polynomial GL-representations, alongside the Schur and Weyl functors, which is of independent interest.

A proper coloring of vertices of a graph $G$ is called $t$-frugal if no color appears more than $t$ times in the neighborhood of any vertex of $G$. While $1$-frugal coloring coincides with $2$-distance coloring, we study $2$-frugal colorings of planar graphs with maximum degree $4$ and $5$, respectively. We prove that every planar graph with maximum degree $4$ admits a $2$-frugal coloring with $7$ colors, while a planar graph with maximum degree $5$ admits a $2$-frugal coloring with $10$ colors. The results present variations of parts of the famous Wegner's conjecture, which for planar graphs with maximum degree $\Delta\in\{4,\ldots,7\}$ states that their $2$-distance chromatic number is at most $\Delta+5$.

The Tu-Deng Conjecture, made in 2011, asserts that for $k\geq2$ and $1\leq t<2^k-1$, the set \[ S_{t,k}=\left\{(a,b)\in\{0,1,\ldots,2^k-2\}^2: a+b\equiv t\pmod{2^k-1},\quad \wt(a)+\wt(b)<k\right\} \] satisfies \[ |S_{t,k}|\leq 2^{k-1}. \] Proofs of the conjecture were given independently by the author and by Liu, Luo and Xie in arXiv in 2026. Let $r=\wt(t)$, and let $z=k-r$ be the number of zeros in the $k$-bit expansion of $t$. We prove that equality holds in the Tu-Deng Conjecture if and only if every cyclic distance between consecutive zeros is at least $z$. We give several other simple necessary and sufficient conditions, including a forbidden-pattern condition, a packing condition for the zero positions, a composition condition for the gaps, and equivalent conditions involving the coefficients of the polynomial $H_W(u,v)$ in the proof of the Tu-Deng conjecture. We also count the equality words having a prescribed number of zeros.

The linear arrangement of a simple graph $G = (V,E)$ is a bijective labeling $\pi$ of the vertices of $G$ to the set $\{1, \ldots, n\}$. The maximum linear arrangement ($\texttt{maxLA}$) problem is to determine whether $G$ has a labeling such that the sum of labels across $G$'s edges, $\sum_{uv \in E} |\pi(u) - \pi(v)|$, is at least a given value $k$. Minimum linear arrangement ($\texttt{minLA}$) is defined analogously. Both $\texttt{maxLA}$ and $\texttt{minLA}$ are known to be NP complete for general graphs. $\texttt{minLA}$ has shown to be NP complete for bipartite graphs by Evans and Shiloah in 1975. It has been an open problem whether the same holds for $\texttt{maxLA}$. In this paper we prove that $\texttt{maxLA}$ is NP complete for bipartite graphs.

We prove that every connected, locally finite, quasi-transitive graph which is quasi-isometric to a graph excluding a countable clique as a minor is quasi-isometric to a planar graph. Consequently, every finitely generated group with such a quasi-isometry is virtually planar, meaning that some finite-index subgroup admits a planar Cayley graph. We also show that a connected, locally finite, quasi-transitive graph admitting a regular map into a bounded-degree graph excluding some finite minor admits a regular map into a bounded-degree planar graph. If the source is accessible, the target need only exclude a countable clique. Combined with work of Mackay, Spriano, and the present author, this shows that a finitely presented group admits a regular map into a bounded-degree graph excluding a countable clique if and only if it is virtually planar.

The Hilton-Milner theorem is a classical theorem in extremal set theory which determines the maximum size of a nontrivial $k$-uniform intersecting family. In this paper, we prove an exact version of the Hilton-Milner theorem in the $\ell_2$-norm. Our method also gives an asymptotic version of the Hilton-Milner theorem in the $(j, p)$-norm.

We prove co-RE-completeness, and thus undecidability, of the following problem: given a single (connected) polycube, decide whether it tiles 3D Euclidean space by translations. We reduce from Wang tiling using the decorated two-prime Sudoku construction of Greenfeld and Tao and a cyclic encoding adapted from OpenAI's 3D aperiodic tile, and apply a reduction of Kim to make the prototile connected (via faces). Dimension three is optimal: translational monotiling is known to be decidable in $\mathbb{Z}^2$ and for a single (possibly disconnected) polyomino in $\mathbb{R}^2$.

We study Gaussian regression under squared population $L_2$ loss in a known $m$-dimensional subspace of degree-at-most-$k$ functions on the $d$-dimensional Boolean cube. Random inputs can undersample regions essential for prediction, delaying the parametric rate even when the model is known. For fixed $q_0<1/2$, $1\le k\le q_0d$, and sufficiently large fixed $A$, the worst-subspace sample threshold for minimax error $A\sigma^2(m+t)/n$ with confidence $1-e^{-t}$, $t\ge\log4$, is \[ N=(m+t)\exp\{E_{d,k}+O(k^{1/3})\}, \quad E_{d,k}=d\Psi(k/d), \] where $\Psi(q)=\log2-\mathsf H(\tfrac12-\sqrt{q(1-q)})$ and $\mathsf H$ is binary entropy with natural logarithms. The upper bound holds for every feasible $m$; the matching lower bound holds when $m\le\binom d{\lfloor k^{1/3}\rfloor}$ or $t\ge m$. We sharpen the Polyanskiy–Samorodnitsky uncertainty principle in two respects. First, for fixed leakage $\rho\in(0,1)$, the smallest set carrying a fraction $1-\rho$ of a nonzero degree-at-most-$k$ polynomial's energy has probability $\exp\{-E_{d,k}+O_{\rho,q_0}(k^{1/3})\}$. An Airy-kernel construction proves that the remainder cannot be $o(k^{1/3})$ in general. Second, we construct a subspace of dimension $\binom d{\lfloor k^{1/3}\rfloor}$ such that every function in the subspace has at least a fraction $1-\rho$ of its energy on the same set, whose probability is at most $\exp\{-E_{d,k}+C_{\rho,q_0}k^{1/3}\}$. For sufficiently large $k$, this set is a Hamming ball. A striking consequence is an exponential cost of noise: the parametric rate can require $(m+t)4^k\exp\{-O(k^{1/3})\}$ samples, whereas $O((m+t)2^k)$ suffice for noiseless identification. As $k\to\infty$ with $k/d\to0$, the noisy threshold is $(m+t)\exp\{2k+o(k)\}$.

We study the Lax operator for Macdonald symmetric functions introduced by Nazarov and Sklyanin in 2015, extending the spectral theory developed by Mickler-Moll in the Jack case. We extend the NS operator to a Lax operator acting on symmetric functions with coefficients in polynomials in an auxiliary variable, and prove it has an orthogonal eigenbasis indexed by partitions and their addable corners, with the corner contents as eigenvalues, and leading terms given by Macdonald symmetric functions. We obtain the Macdonald analogue of the author's earlier Jack box-sum identity for Littlewood-Richardson coefficients. We then strengthen this box-sum identity to a two-parameter formula, proved using a finite-rank product formula of Warnaar. Its residues give explicit linear relations among Macdonald Littlewood-Richardson coefficients.

Due to the availability of powerful analytic techniques, vastly superior lower bounds are known for the measurable chromatic number of Euclidean spaces compared to their ordinary chromatic number. Indeed, even the breakthrough lower bound of 5 for the famous Hadwiger-Nelson problem lagged over 35 years behind that of the measurable setting. This raises the fundamental question of whether the measurable and ordinary chromatic number of Euclidean spaces differ as conjectured by Székely in 1984. Our main result is that $\overline\alpha(\mathbb{R}^4)=m_1(\mathbb{R}^4)$ and $\chi(\mathbb{R}^4)=\chi^{(m)}(\mathbb{R}^4)$, and for $d\ge5$ that \[ \overline\alpha(\mathbb{Q}^d) = \overline\alpha(\mathbb{R}^d)=m_1(\mathbb{R}^d) \qquad\text{and}\qquad \chi(\mathbb{Q}^d) = \chi(\mathbb{R}^d)=\chi^{(m)}(\mathbb{R}^d). \] Our theorem also holds for multiple forbidden distances $D=\{d_1,\ldots,d_t\}$ provided that $d_1^2,\ldots,d_t^2 \in \mathbb{Q}$. As a consequence, we immediately lift numerous measurable chromatic number results into the ordinary setting. We also take the opportunity to further optimize the new bounds. For multiple distances, Erdős asked whether the chromatic number of $\mathbb{R}^d$ with up to $k$ forbidden distances $D$ grows exponentially in $k$. By a theorem of Bukh, we obtain for $d \ge 4$ that \[ \sup_{|D|=k}\chi_D(\mathbb{R}^d) \ge m_1(\mathbb{R}^d)^{-k}. \] Making progress on another problem of Erdős, we prove that \[ (2+o(1))^d \le \chi(\mathbb{R}^d) \le \left(\frac{3\sqrt{3}}{4}+o(1)\right)^d. \] We also vastly improve the lower bounds for $\chi(\mathbb{R}^d)$ for small $d\ge4$. We expect that our techniques could be developed much further. This includes the possibility of extending our main theorem that $\chi(\mathbb{R}^d)=\chi^{(m)}(\mathbb{R}^d)$ for $d\ge 4$ to $d=3$ or possibly even to $d=2$ to tackle the Hadwiger-Nelson problem.

We establish a sufficient condition for infinite log-concavity and apply it to five families of combinatorial sequences. We give a new proof of infinite log-concavity for the Boros–Moll coefficient sequences $(d_k(n))_{k=0}^n$ for fixed $n$. For the transposed sequences $(d_\ell(\ell+k))_{k\ge0}$, we prove Zhao's conjecture on infinite log-concavity for every fixed integer $\ell\ge3$. For the normalized sequences in Euler's difference table, we confirm the infinite log-concavity conjecture of Chen, Gu, Ma and Wang under essential iteration, in which both endpoints are discarded after each step. We also confirm a conjecture of Medina, Moll and Rowland on the infinite log-concavity of the coefficient sequences of polynomials arising from iterated primitives of $\log(1+x)$. Finally, we answer a question of Brändén and Chasse by showing that, for positive integers $d$, the sequence $(k^d)_{k\ge0}$ is infinitely log-concave if and only if $d\ne2$. For every real $d\ge3$, all its iterates are positive at every index $k\ge1$. The proofs combine holomorphic estimates with finite computer-assisted verification.

We study arrangements of conics and lines in the complex projective plane from the viewpoint of 4-manifolds. We introduce combinatorial conic-line arrangements, the conic-line counterpart of rank-3 matroids, together with their topological and smooth realisations, and we ask which restrictions on a complex arrangement survive when its components are replaced by locally-flat spheres in the same homology classes with the same local singularity models. For an ADE combinatorial conic-line arrangement of even total degree admitting a topological realisation we prove a Hirzebruch-type inequality; the proof uses only branched covers and the local topology of rational double points. In the complex-algebraic category we show that equality holds exactly when the arrangement is a maximising curve, hence free with prescribed exponents. We then introduce divisible and odd combinatorial conic-line arrangements and obtain inequalities and congruences for the multiplicity sequence of their resolutions, via cyclic branched covers, the G-signature theorem, spin structures, and Furuta's 10/8-theorem.

Which pure states give uniform outcomes in several mutually unbiased measurements, and when can they form another measurement basis? We distinguish common unbiased states, their orthonormal completion, and further extension. For a displayed phase-deformed Fourier pair in dimension $2m$, a clock symmetry and skew antiunitary complete every prescribed common unbiased state to an explicit orthonormal basis. In dimension six, an amplitude-profile obstruction proves strong unextendibility of a given mutually unbiased triple when its original third-basis rays are invariant under the ternary clock. We give an exact nonproduct example and an application to a companion whose columns are localized at single Fourier frequencies. For every exact Hadamard matrix in a specified closed entrywise neighborhood of a fixed reference matrix, the coordinate and Hadamard bases have exactly 48 common unbiased rays but no third measurement basis, with a quantitative lower bound on the squared Gram defect of every six-column candidate. We also study auxiliary complex equations for common states, retaining algebraic multiplicities. For a flat unitary of order six with all $3\times3$ minors nonzero, the projective ray scheme is finite; each singular two-minor contributes two reduced isolated boundary points, and the affine coordinate algebra is finite-dimensional over $\mathbb{C}$. In every order, a critical-algebra identity preserves multiplicities; in dimension six it gives finiteness for a fixed phase fibre. Specialization over a discrete valuation ring controls complete reciprocal-orthogonal collections. A nonphysical Tao/Potts collection separates algebraic from physical solutions. These results do not determine the global maximum number of mutually unbiased bases or classify all triples.