Papers

For a finite sequence of positive integers $\boldsymbol{a}=(a_1,\dots,a_n)$, the restricted partition function $q_{\boldsymbol{a}}(k)$ denote the number of nonnegative integer solutions to the equation $a_1x_1+a_2x_2+\cdots +a_nx_n=k$. It is proved to be a quasi-polynomial of degree $n-1$. Write $q_{\boldsymbol{a}}(k)=\sum_{j=0}^{n-1}c_j(k)k^j$ with periodic coefficient functions $c_j$, and set $b_m=\#\{i:m\mid a_i\}$. In 2008, Beck, Sam, and Woods conjectured that the minimum period of $c_j(k)$ is $\mathrm{lcm}\{m:b_m>j\}$. In this paper, we derive an exact root-of-unity formula for every coefficient function $c_j(k)$. The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of $c_j(k)$. Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.

Let $\mathcal H=(V,E)$ be a locally finite simple hypergraph, equip $V$ with the hyperpath metric, and consider the equal-edges random walk \cite{CoupetteEtAl2023}. For adjacent vertices $x$ and $y$, we prove that the idleness function $\alpha\mapsto\kappa_\alpha^{\mathcal H}(x,y)$ is piecewise affine and with no more than three affine pieces. A separate mass-balance argument gives linearity on $[1/2,1]$ for every locally finite simple hypergraph and, consequently, a limit-free expression for the Lin–Lu–Yau curvature. In the $r$-uniform linear case, the hypergraph walk agrees exactly with the simple random walk on its 2-section. This reduction transfers the sharp endpoint intervals of Bourne, Cushing, Liu, Münch, and Peyerimhoff \cite{BourneEtAl2018}.

Let $R_r(k)$ denote the diagonal $r$-color graph Ramsey number. We prove that there exist absolute constants $c,K>0$ such that \[ R_r(k)\le \exp\!\left(-c\frac{k}{r^2\log^4(2r)}\right)r^{rk} \] for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. The proof combines a positive-coefficient root filter of variable order with a retained-spine refinement of the multicolor book method.b

Discrete Forman-Ricci curvature is a quantity associated to each edge of a graph that describes its local geometry. It has proven to be a useful tool in network analysis in a variety of applications. Recent work by Roost et al.\ (2024) proposed the use of Markov bases to sample from the space of graphs with prescribed vertex degrees and curvatures. In the present work, we further develop the algebraic and combinatorial theory of these Markov bases. We show that the degree of an indispensable Markov move grows at least quadratically in the maximum degree of the graph. In light of this result, a compact description of all Markov basis elements seems unattainable at present. Instead, we find a lattice basis for this problem using only degree three Markov moves, which allows us to employ recently-developed reinforcement learning methods for finding Markov moves that can be applied to a specific graph.

The \emph{$k$-color Ramsey number} $R_k(C_{2\ell+1})$ is the least integer $n$ such that any $k$-edge-coloring of a complete graph $K_n$ has a monochromatic odd cycle $C_{2\ell+1}$. Axenovich, Cames van Batenburg, Janzer, Michel, and Rundström~(JCT-B, 2026) recently proved \[ R_k(C_{2\ell+1})\le (4\ell-2)^k k^{k/\ell}+1, \] and Miyazaki, Mulrenin, Pohoata, and Zheng further improved the factor $k^{k/\ell}$ to $(k!)^{1/\ell}$. As Jenssen and Skokan (AM, 2021) determined $R_k(C_{2\ell+1})$ for fixed $k$ and sufficiently large $\ell$, it becomes even more interesting to seek better bound for fixed $\ell$ and sufficiently large $k$. In this paper, we show \[ R_k(C_{2\ell+1}) \le \frac{2\ell}{2\ell-1}(2\ell-1)^k(k!)^{1/\ell} \exp\!\left(k^{1-1/\ell}+O_\ell\!\left(k^{1-2/\ell}+\log k\right)\right)+1 \] for every fixed $\ell\ge 2$ and sufficiently large $k$, which improves the bound of Miyazaki et al. by a factor $2^{k-o(k)}$, and the bound of Axenovich et al. by a factor $(2\e^{1/\ell})^{k-o(k)}$.

Let $\Gamma=(G,\sigma)$ be a signed graph, where $G$ is the underlying graph with vertex set $V(G)$ and edge set $E(G)$ such that $\sigma: E(G)\to \{-1,1\}$ is the sign function. For $U\subset V(G)$, the operation that changes the sign of all edges between $U$ and $V(G)\setminus U$ is called switching. Two signed graphs with the same underlying graph are switching equivalent if one is obtainable from the other one by switching a subset. Two signed graphs are switching isomorphic if one is isomorphic to a switching equivalent signed graph of the other one. A signed cycle is called negative if it contains an odd number of negative edges. A signed graph is balanced if none of its cycles is negative; otherwise it is unbalanced. The adjacency matrix $A(\Gamma)$ of $\Gamma$ is obtained from the standard $(0,1)$-adjacency matrix of $G$ by reversing the sign of all $1$s which correspond to negative edges. The index of $\Gamma$ is the largest eigenvalue of $A(\Gamma)$ and the spectral radius of $\Gamma$ is the largest absolute value of the eigenvalue of $A(\Gamma)$. The least eigenvalue of $\Gamma$ is the least eigenvalue of $A(\Gamma)$. We study the extremal problems of the index and the spectral radius among unbalanced signed multipartite graphs. More precisely, we determine the unbalanced signed $t$-partite graphs with fixed $t\ge 2$ and partite sizes (order, respectively) that maximizes the index and the spectral radius respectively, up to switching isomorphism. To determine the unbalanced signed multipartite graphs with fixed partite sizes (order, respectively) with maximum spectral radius, we also determine those with minimum least eigenvalue.

The bipartite Turán number of a graph $H$, denoted $\text{ex}(m, n; H)$, is the maximum number of edges in any $H$-free bipartite graph $G = (A, B; E)$ with parts of size $|A| = m$ and $|B| = n$. We study this problem for two families. For a tree $T = T(r, s)$ with parts $R$ and $S$ of sizes $|R| = r \le s = |S|$, we prove \[ (r - 1) n \;\le\; \text{ex}(m, n; T(r, s)) \;\le\; (r - 1) n + C(m,r,s) \] for $n$ sufficiently large compared to $m$, $r$, and $s$, where $C(m,r,s)$ does not depend on $n$, determining the leading-order term exactly (with the star case $r = 1$ solved with an exact formula). For a star forest $F = \bigcup_{i=1}^k S_{d_i}$ with $d_1 \ge \cdots \ge d_k$, we determine the exact value $\text{ex}(m, n; F) = (k - 1) n + (d_k - 1)(m - k + 1)$ for $n$ sufficiently large, and characterize the unique extremal graph.

We present a construction of a family of $0/1$-polytopes whose edge expansion decreases exponentially with the dimension, which disproves the Mihail-Vazirani conjecture that the graph of every $0/1$-polytope has edge expansion at least one.

For a prime \(p\), let \(A_p=\{k!\pmod p:1\leq k<p\}\). We prove \(|A_p|\gg p^{8/15}\), improving the general lower bound \((\sqrt{2}-o(1))p^{1/2}\). The proof begins with the identity \((n+2)!=(n+1)!+((n+1)!)^2/n!\) in \(\mathbb{F}_p\), which produces many incidences for a family of fractional-linear maps. After Cauchy–Schwarz, the transition maps between two members of this family become affine lines, with multiplicity at most two. The Cartesian-product point-line incidence theorem of Stevens and de Zeeuw then yields the exponent \(8/15\).

The partial Petrial polynomial of a bouquet can be computed from the coranks over GF(2) of matrices obtained by varying the diagonal entries of the adjacency matrix of its intersection graph. Motivated by this matrix formulation, we study the corresponding polynomial for simple graphs and determine when it has exactly two or three nonzero terms. A key tool is the interpolating property: the exponents of the nonzero terms are consecutive. Using this property together with local complementation minors of grafts, we extend the known characterization of the binomial case from connected circle graphs to all connected simple graphs, showing that such a graph has a binomial partial Petrial polynomial if and only if it is a path. Our main result characterizes the trinomial case for trees: a tree has a trinomial partial Petrial polynomial if and only if it is a T-shape tree or an H-shape tree. Here, a T-shape tree has maximum degree 3 and exactly one vertex of degree 3, whereas an H-shape tree has maximum degree 3 and exactly two vertices of degree 3, which are adjacent. We also derive explicit formulas for both families in terms of Jacobsthal numbers.

We introduce a Lovász-type semidefinite parameter for balanced colouring of signed graphs. Its homomorphism target is a unit sphere equipped with an orthogonal involution: the fixed and anti-fixed components play different roles, while applying the involution to one endpoint realizes switching. The resulting parameter admits a symmetric formulation with two positive semidefinite matrices and an equally symmetric dual. It is also one half of the strict vector chromatic number of the ordinary graph formed by the negative edges of the double switching graph. Our second main contribution is a hierarchy of convex corners. Starting from the balanced induced subgraph polytope, we define signed analogues of the stable-set, theta, and clique-inequalities relaxations in the original vertex space. The signed theta body has an intrinsic two-matrix description, its all-ones gauge is the new scalar parameter, and the ordinary stable-set hierarchy is recovered exactly from signed digon graphs. For all-negative signatures, the construction becomes a relaxation of the maximum induced bipartite subgraph problem and is related to the generalized theta number. Finally, we propose a notion of balanced perfectness and show that it is strictly weaker than perfectness of the associated double-cover graph.

Let $K$ be a finite $k$-dimensional simplicial complex with $k\ge1$, and let $\Lup_{k-1}(K)$ be its $(k-1)$-dimensional up-Laplacian. We resolve the equality case of a conjecture of Duval and Reiner: for a pure complex, the nonzero spectrum of $\Lup_{k-1}(K)$ equals ${d^{\mathrm v}(K)}^{\T}$, the conjugate vertex-degree partition, if and only if the complex is isomorphic to a shifted complex. In fact, we prove more: equality of the second power sums alone already forces shiftedness, and the gap between them equals twice the number of failed elementary shifts of facets. We next characterize equality in the Duval–Reiner bound $\lambda_1(\Lup_{k-1}(K))\ge d_1(K)+k$, where $d_1(K)$ is the maximum upper degree of a ridge. We introduce the clean ridge, a higher-dimensional analogue of a dominating vertex, and combine it with a signed Sachs expansion for the associated signed facet-adjacency matrix to obtain a necessary and sufficient combinatorial criterion. When the signed facet-adjacency graph is connected and balanced—with balance equivalent to disorientability in dimension $k$—the criterion reduces to the simple condition that all facets share a common ridge. Finally, whenever $K$ has at least two facets, we prove the sharp bound \[ \lambda_2\bigl(\Lup_{k-1}(K)\bigr)\ge d_2(K)+k-1. \] This gives a higher-dimensional analogue of the second-index Brouwer–Haemers bound for graphs. The natural termwise extension to higher eigenvalue indices fails already at $m=3$, as a s

Erdős Problem 725 asks for an asymptotic formula for the number $L_{k,n}$ of ordered, labelled $k\times n$ Latin rectangles. Godsil and McKay proved that $L_{k,n}\sim (n!)^k((n)_k/n^k)^n(1-k/n)^{-n/2}e^{-k/2}$ for $k=o(n^{6/7})$. We provide a partial solution to Erdős Problem 725 by proving this asymptotic for every $k=o(n)$. More precisely, set $\widetilde A_{k,n}=(n!)^k((n)_k/n^k)^n\exp\{[n(H_n-H_{n-k})-k]/2\}$. For every $K(n)=o(n)$, uniformly for $0\leq k\leq K(n)$, we prove $\log(L_{k,n}/\widetilde A_{k,n})=O(k^2/n^2)$, with an absolute implied constant. The results of this paper have been formally verified in Lean.

The Zarankiewicz number z(n;2) is the largest number of ones in an n x n zero-one matrix with no all-one 2x2 submatrix. At n=43, the Kovari-Sos-Turan bound, sharpened by Reiman, gives z(43;2) <= 301 with no rounding slack; equality would force a projective plane of order six, which Tarry ruled out in 1900. That fact alone says nothing about how far below 301 the truth lies. We prove z(43;2) <= 294. The argument is local and geometric. Writing D = 301 - E for the deficiency of a configuration with E ones, a crossing count paired with its dual forces every line to at most eight points and every point to at most eight lines whenever D <= 6. A deficiency count then produces a clean point of degree seven, collinear with all others, unless the configuration has both a line of eight points and a point of degree eight. The seven lines through a clean point partition the remaining 42 points into groups that every other line meets at most once, so the number of (line, missed group) incidences equals D. At D=6 this forces group sizes 6,6,6,6,6,6,6 or 7,6,6,6,6,6,5; the second forces a TD(4,6), and the first yields 32 blocks of a partial TD(5,6), whose refutation needs an unknown packing number. We determine it. pa(5;6) and pa(6;6) are listed as open in the Handbook of Combinatorial Designs at 30 <= N <= 34; a class holds at most six words, so at most one full class gives at most 6 + 5*5 = 31, and what was missing was refuting the two-full-class case, done here by exhaustive search. Hence pa(5;6) = pa(6;6) = 31. With no clean point, the configuration is rigid enough to complete to a projective plane of order six. With a known 290-one configuration, this gives 290 <= z(43;2) <= 294. We also derive z(43;2) <= 299 via Totten's classification, and rule out configurations with 290+ ones from deleting rows/columns of PG(2,7).

A set of vertices $S$ of a graph $G$ is locating-dominating if $S$ is dominating and, for each pair of distinct vertices not in $S$, their neighborhoods in $S$ are distinct. We present results on the minimum density of such sets in the infinite hexagonal grid with a finite number of rows $k$, also known as the hexagonal strip of width $k$, which we denote by $H_k$. For each $k\geq 2$, we present either an optimal solution or a quasi-optimal solution for $H_k$ that is within $1.3\%$ of the optimum. We describe an exact exponential-time algorithm for fixed k, which we implemented to find optimal solutions for $k \leq 5$. As the infinite grid $H_{k}$ always admits a periodic optimal solution, to deal with larger values of $k$, we present an integer linear program that finds an optimal periodic solution for $H_{k}$ for each fixed period. This program yields high-quality feasible solutions for $H_7$ and $H_8$, which we then combine with an optimal solution for $H_3$ to obtain quasi-optimal solutions for all $k\geq 6$. All these solutions admit a very short description.

Put $q=r-1$, $t=k-1$, and $D=tq+1$. For an edge $e$ of a linear $C^r_{1,k}$-free $r$-uniform hypergraph, define \[ \delta_H(e)=\sum_{v\in e}\frac{1}{d_H(v)}-\frac{r}{D}. \] The defect satisfies $\delta_H(e)\ge 0$. At equality, every vertex of $e$ has degree $D$, and the petal trace at $e$ is a disjoint union of $t$ affine planes of order $q$. Equivalently, restoring the base line gives $t$ projective planes of order $q$ with common line $e$. For growing crowns, the equality structure is stable in the following sense. If $q_j\to\infty$, $2\le t_j\le q_j$, and $e_j$ is an edge of a finite linear $C_{1,t_j+1}^{q_j+1}$-free hypergraph satisfying \[ \frac{t_j^2}{q_j}\to0, \qquad t_j^3\delta_{H_j}(e_j)\to0, \] then, for every fixed $0<\theta<1$, \[ \frac{ |\{f\ne e_j:f\cap e_j\ne\varnothing,\ \delta_{H_j}(f)\ge\theta/t_j^2\}| }{(q_j+1)(t_jq_j)} \to1. \] Thus an edge close to equality is adjacent almost entirely to edges with defect of order at least $t_j^{-2}$. A uniform form gives absolute constants $Q_0,\varepsilon_0,c_0>0$ such that, whenever $q\ge Q_0t^2$ and $\delta_H(e)<\varepsilon_0/t^3$, at least $\tfrac12(q+1)tq$ neighbors of $e$ have defect at least $1/(100t^2)$. Consequently, \[ c^{\mathrm{lin}}_{q+1,t+1}\le t-\frac{c_0}{t}, \] where $c^{\mathrm{lin}}_{r,k}$ denotes the asymptotic linear Turán coefficient for $C^r_{1,k}$.

A certified exhaustive computation shows that every simple cubic bipartite graph on at most 58 vertices contains a cycle of length 4, 8, or 16. Consequently, any cubic bipartite counterexample to the Erdos-Gyarfas conjecture has at least 60 vertices, improving the established published lower bound for this class from 30 to 60. The proof begins with a Moore-bound observation: below 62 vertices, a cubic bipartite graph avoiding 4- and 8-cycles must contain a 6-cycle. Viewing the graph as the Levi graph of a linear symmetric v3-configuration turns this 6-cycle into a Berge triangle. Up to symmetry, only two rooted extensions are possible. A complete restricted-growth search on at most 29 points exhausts both search trees. The computation is checked by two separately implemented exact procedures using different C16 oracles and by a static witness certificate. Source code, certificates, and reproduction instructions are archived with the paper.

For a positive integer $n$, let $\mathcal{M}_n$ be the set of distinct multinomial coefficient values $n!/(p_1!\cdots p_t!)$, where $(p_1,\ldots,p_t)$ ranges over the integer partitions of $n$. We study two complementary aspects of the additive structure of $\mathcal{M}_n$: the maximum cardinality $s(n)$ of a Sidon subset and the multiplicities of unordered pair sums in the full set. A product embedding, strongly Sidon subsets, and estimates for prime partitions give \[ \liminf_{n\to\infty}\frac{\log(s(n)/n)\log\log n}{\sqrt{\log n}}\geq\frac{\pi}{\sqrt{3}}. \] Long arithmetic progressions and separated copies yield a complementary lower bound $2/(3\sqrt{3})$ for the normalized Sidon defect. Writing $r_n(t)$ for the number of unordered representations $t=x+y$ with $x,y\in\mathcal{M}_n$, we study the collision excess $C(n)$, the number $D(n)$ of multiply represented sums, the maximum multiplicity $\mu(n)$, and the cumulative profile $T(n,k)$. We prove \[ \liminf\frac{C(n)}{n^2\log n}\geq\frac18,\qquad \liminf\frac{D(n)}{n^{3/2}\sqrt{\log n}}\geq\frac{4}{3\sqrt{3}},\qquad \liminf\frac{\mu(n)}{\sqrt{n\log n}}\geq\frac12. \] We also obtain a scaled lower envelope for the full profile, an additive-energy bound, stabilization with respect to the number of variables, and an explicit family of trinomial collisions. Exact Sidon values through $n=16$, the lower bound $s(17)\geq89$, and pair-sum statistics through $n=20$ are reported with reproducible computational material.

A key theme in modern extremal combinatorics is the study of classical combinatorial theorems relative to sparse subsets of their natural settings. Here we describe some of the recent progress in this area and state a number of problems that remain open and pressing.

We give a quantitative combinatorial characterization of size-oblivious one-sided testability in the dense graph model, resolving a question of Alon, Fischer, Newman, and Shapira. For hereditary graph properties, we prove that one-sided testability is quantitatively equivalent to the existence of suitable hypergraph containers, a central and widely used tool in modern combinatorics. Combining this equivalence with the Alon-Shapira notion of semi-hereditariness yields a quantitative characterization of arbitrary graph properties. The correspondence is effective in both directions and provides explicit translations between tester complexity and container parameters. Our proof is regularity-free and extends uniformly to every fixed finite relational signature of bounded arity, including digraphs, coloured graphs, and hypergraphs. As applications, we obtain quantitative closure results for partition properties and testers for properties defined by the existence of a linearly large induced substructure.

Let $G$ be a (multiplicatively written) finite group. A sequence over $G$ is a finite collection of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose terms can be ordered such that their product in $G$ equals the identity element of $G$. The set $\mathcal B (G)$ of all product-one sequences over $G$, endowed with the concatenation of sequences as the operation, is a finitely generated C-monoid; in particular, it is atomic, i.e., every non-unit element can be written as a finite product of atoms. The study of $\mathcal B (G)$ is of fundamental importance, as its combinatorial, algebraic, and arithmetic properties play a crucial role across various branches of mathematics, most notably in invariant theory and factorization theory. While the arithmetic of the monoid $\mathcal B (G)$ is well understood in the abelian setting (in which case $\mathcal B (G)$ is a Krull monoid), little is known in the non-abelian setting because of the substantial structural complexity involved. In this paper, we study the arithmetic invariants of the monoid $\mathcal B (G)$ for non-abelian groups, focusing in particular on the catenary degree. The catenary degree $\mathsf c (G)$ of the monoid $\mathcal B (G)$ is defined as the smallest integer $N$ such that any two factorizations of an element $S \in \mathcal B (G)$ can be concatenated by a chain of factorizations in which adjacent steps differ by replacing at most $N$ atoms. Extending the methods from arithmetic combinatorics to the non-abelian setting, we explicitly characterize all finite groups with catenary degree at most 3, and we investigate an infinite class of finite groups whose monoids of product-one sequences are seminormal and possess well-behaved arithmetic structures. Furthermore, we show that a specific non-abelian group in this class has catenary degree 4.

We introduce a special mapping from pairs of power series to the group of Sprugnoli matrices. This mapping has the property when the second argument is an even power series, then the square of the resulting Sprugnoli array is an aerated element of the Appell subgroup of the Riordan group. This allows us to explore the square roots of elements in the aerated Appell subgroup. As the identity is an element of this subgroup, we are led to explore related involutions in the Sprugnoli group.

We consider degenerate principal series representations $\operatorname{Ind}_P^G\chi$ over finite fields, where $G$ is a classical subgroup of $\operatorname{GL}_{2n}$, and $P$ is the Siegel parabolic subgroup. For example, we show that this representation is always multiplicity-free and irreducible for generic characters $\chi$. We then discuss a particular intertwining operator $I$ on $\operatorname{Ind}_P^G\chi$ and its related combinatorics. Firstly, this operator $I$ produces families of diagonalizable antitriangular matrices with well-behaved eigenvalues. Secondly, applying $I$ to a special vector in $\operatorname{Ind}_P^G\chi$ leads us to various matrix Gauss sums, whose evaluations imply an explicit equidistribution result of the trace and determinant of symmetric and alternating invertible matrices.

We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first $N$ orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we prove that on a flat $d$-dimensional torus the number of such distances is at most $2^d+1$. The main tool is a new growth theorem for the denominators $q_1<q_2<\cdots$ of best simultaneous approximations in a $d$-dimensional inner-product space, which is of independent interest. We prove that, whenever $q_{n+2^d}$ is defined, either $q_{n+2^d}\ge2q_{n+1}$, or the indices $1,\ldots,2^d$ can be partitioned into disjoint pairs $\{j,k\}$, $j<k$, such that $q_{n+k}=q_n+q_{n+j}$. In particular, $$ q_{n+2^d}\ge \min\{2q_{n+1},q_n+q_{n+2^{d-1}}\}\ge q_n+q_{n+1}. $$

The X-ray of a permutation is a sequence of sums along each diagonal of the associated permutation matrix. They satisfy certain necessary constraints on distribution of the values, which are conjectured to be sufficient when the sequence is binary. By re-expressing the constraints in a form that allows for real-valued relaxations, we prove that these binary sequences are always X-rays of doubly stochastic matrices.

For a finite simple graph $G$, let $t(G)$ be the largest order of an induced tree and let $g(G)$ be the girth. We prove three consecutive conjectures of DeLaViña's Graffiti.pc program. First, writing $\ell(v)$ for the independence number of the subgraph induced by the neighbourhood of $v$, we prove $t(G) \ge \lfloor g(G)/2 \rfloor - 1 + \max_{v \in V(G)} \ell(v)$. Second, if $\mathrm{Per}(G)$ is the periphery and $f(G) = \max_x d(x, \mathrm{Per}(G))$, we prove $t(G) \ge \frac{2}{3} g(G) + f(G)$, and establish the stronger integral bound $t(G) \ge f(G) + \lceil 2g(G)/3 \rceil$ when $G$ contains a cycle. Third, if $\delta'(G)$ is the second-smallest degree, counted with multiplicity, then every connected non-tree graph satisfies $t(G) \delta'(G) \ge g(G) + 1$. These are Conjectures 141, 142, and 143 of Written on the Wall II. Complete, machine-checked Lean 4 proofs of all three formal statements accompany the manuscript.

We prove the middle-stair conjecture for every complete bipartite graph. If a parallel chip-firing game on $K_{a,b}$ has configuration $\sigma$ with $2ab-a-b<|\sigma|<2ab$, then its eventual period is $2$. The balanced case $K_{a,a}$ was proved by Ji, Li, and Wang using one-parameter conjugate configurations. We introduce two-parameter conjugates $c^{k,\ell}$, in which the rank shift on one side supplies the additive offset on the other. These conjugates preserve both the total number of chips and the activity. An exact Ferrers-diagram count then produces a nonnegative conjugate with two-round firing coverage on one side. The coverage propagates in alternating two-round waves, giving activity $1/2$; non-clumpiness then forces period $2$.

We classify freeness for finite central arrangements whose underlying matroids are regular. Let $\mathcal A$ be such an arrangement over an arbitrary field, and put $M=M(\mathcal A)$. Then $\mathcal A$ is free if and only if $M$ is supersolvable; equivalently, $M$ admits a nice partition; equivalently, $M$ is the cycle matroid of a chordal simple graph. Thus, for arrangements with regular underlying matroids, freeness has a complete combinatorial classification independent of the base field. We use Seymour's decomposition theorem for regular matroids to prove that freeness forces supersolvability. We also characterize nice partitions of finite simple binary matroids: a partition is nice if and only if it is independent and no line is contained in a single block. Consequently, a finite loopless binary matroid admits a nice partition if and only if it is simple and supersolvable.

Let $D(N)$ denote the largest cardinality of a subset of $\{1,\ldots,N\}$ containing no nonzero square difference. While a construction certifying $D(N)\geq (1-o(1))N^{1/2}$ is almost trivial, Erdős conjectured that this bound is sharp up to polylogarithmic factors. This was disproved by Sárközy and later again by Ruzsa, who found an elegant construction showing that $D(N)\geq c\cdot N^{0.733077\dots}$, with an absolute constant $c>0$. His approach was subsequently refined, leading to the previously best known lower bound with exponent $0.7334117\dots$ due to Beigel-Gasarch and, independently, Lewko. However, in the original paper Ruzsa observed that $3/4$ seems to be the natural barrier of his approach. In this paper we develop a new construction leading to the lower bound \[ \liminf_{N\to\infty}\frac{\log D(N)}{\log N} \geq \alpha_*:= 0.7527964558\ldots; \] thus crossing the natural exponent-$3/4$ barrier of Ruzsa's method. The value $0.7527964558\ldots$ arises from a simple optimisation problem and appears to be the limit of the new approach.

In this paper, we study the distribution of simplices in both discrete and continuous settings. Let $q$ be an odd prime power, let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^d$, and let $2\leq k\leq d-1$. We prove that every set $E\subset\mathbb F_q^d$ with \[ |E|\geq C_{d,k}q^{\beta_{d,k}}, \qquad \beta_{d,k}= \begin{cases} \displaystyle \frac{d+k}{2}-\frac{k-1}{k+1}, & d-k\ \text{even},\\[2mm] \displaystyle \frac{d+k-1}{2}, & d-k\ \text{odd}, \end{cases} \] determines a positive proportion of all ordered nondegenerate $k$-simplex congruence classes. This improves the previous exponent due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2017), and is sharp when $d-k$ is odd. In the Euclidean setting, we prove that if $E\subset\mathbb R^d$ is compact and $\dim_{\mathrm H}(E)>d-1$, then there exists a Frostman probability measure $\mu$, supported on $E$, and a set of pins of full $\mu$-measure such that the pinned distance configuration measure for labeled $(d-1)$-simplices is absolutely continuous at every such pin. We also show that the same conclusion holds when $E\subset\mathbb R^d$ is a compact Salem set with $\dim_{\mathrm H}(E)>k$.