Papers

For a line arrangement $\mathcal{A} \colon f=0$ in $\mathbb{P}^2$, let $\nu(\mathcal{A})$ be the maximal dimension of a graded piece of the Jacobian module of $f$. We study the conjecture that $\nu(\mathcal{A})$ depends only on the intersection lattice of $\mathcal{A}$. It is known that $\mathcal{A}$ is free if and only if $\nu(\mathcal{A})=0$, and hence this conjecture is a strengthening of Terao's conjecture for line arrangements. We show that the conjecture holds for arrangements of at most $13$ lines, except possibly for arrangements of exactly $13$ lines whose intersection points have maximal multiplicity $5$.

The inducibility problem asks for the maximum number of induced copies of a fixed graph among all graphs with a prescribed number of vertices. Inducibility has been an active area of research in extremal combinatorics, but determining all extremal graphs for explicitly defined patterns remains challenging, particularly when the description is required to hold at every host order. A graph $H$ is called a fractalizer if, for every positive integer $n$, every $n$-vertex graph maximizing the number of induced copies of $H$ is a balanced iterated blow-up of $H$, obtained by recursively repeating the same pattern in parts whose sizes differ by at most one. Previous probabilistic results show that large random graphs and random abelian Cayley graphs are fractalizers with probability tending to one, establishing their abundance without directly providing explicit families. We prove that every sufficiently large prime-order Paley graph is a fractalizer. Thus these classical arithmetic patterns determine the exact recursive structure of every extremal host, at every host order and without any algebraic assumptions on the host. We also construct a second explicit infinite family of nontrivial Cayley fractalizers, for which the fractalizer property admits a simpler proof than in the Paley case. Together, these results resolve the explicit-construction question discussed at the 2025 American Institute of Mathematics workshop "Flag Algebras and Extremal Combinatorics."

We introduce and study Reed–Solomon sunflowers, namely families of Reed–Solomon codes whose pairwise intersections are all equal to the same fixed subspace. This notion lies at the intersection of extremal subspace combinatorics and coding theory: it can be viewed as a structured version of the sunflower problem in the Grassmannian, and it naturally produces constant-dimension subspace codes with prescribed minimum distance. We focus mainly on the case in which the center is the one-dimensional space generated by the all-one vector. We give an algebraic criterion, expressed in terms of generalized $V$-matrices, ensuring that a family of Reed–Solomon codes forms such a sunflower. We then study the size of these families through counting and constructions. In dimension two, we show that all distinct Reed–Solomon codes form a sunflower and determine its size by counting Reed–Solomon codes up to affine equivalence of their evaluation vectors. For fixed dimension $k\geq3$ and length $\ell\geq2k-1$, we give an explicit recursive construction with $\Omega_{k,\ell}(q^{\lfloor\ell/(2k-1)\rfloor})$ petals and a greedy existence argument with $\Omega_{k,\ell}(q^{\ell-2k+2})$ petals as $q\to\infty$. We also apply the greedy argument to obtain families of $[\ell,k]_q$ MDS codes of size $\Omega_{k,\ell}(q^{2(\ell-2k+2)})$, whose pairwise intersections have dimension at most one but need not be equal.

We prove the conjecture of Laubie on dimensions of components of two operads completing Loday's operadic butterfly diagram. For one of these operads, the dimension of the $n$-th component is equal to $\binom{2n}{n-1}$ for all $n\ne 3$, and for the other it is equal to $\binom{2n}{n-1}+1$ for all $n\ge 5$.

Let $\Delta_u(d,n)$ be the maximum graph diameter of a pointed $d$-dimensional polyhedron with $n$ facets. Using a curved-path encoding of the iterated Kalai–Kleitman recurrence, we prove, uniformly over $n\geq d\geq 4$, \[ \Delta_u(d,n) \leq (n-d)^{\log_2 G_d},\qquad G_d = (4\ln 2+o(1))\frac{d}{(\ln d)^2}, \] where the asymptotic expression for $G_d$ is understood as $d\to\infty$, improving the exponent of the previous best quasi-polynomial bound by an additional logarithmic factor. With the quantitative $d$-step reduction, we also obtain the complementary excess-based bound \[ \Delta_u(d,n) \leq (n-d)^{\frac{1}{2}\log_2(n-d)+O(1)}, \] where the implied constant is absolute. In the regime $n - d = \Theta(d)$, the latter bound is asymptotically stronger and halves the leading coefficient in the exponent. We further examine their behavior as $n$ grows relative to $d$, obtaining sharper exponents when $n = d^{1/\gamma+o(1)}$ for fixed $0<\gamma<1$ and an almost-linear bound in the deep-tail regime $(\ln n)/d\to\infty$. We also show that the leading term of the general bound is sharp within this positive path-counting framework.

We study the sequence $a(0)=0$, $a(1)=1$ and $a(n)=n-a(n-a(n-a(n-1)))$ for $n\ge 2$, listed as A076502 in the On-Line Encyclopedia of Integer Sequences. We identify $a(n)$ as a two-position shift in the greedy Padovan numeration system, with a finite-state correction. The proof constructs an addition automaton from an exact integer-carry invariant and certifies its completeness by finite-language inclusion; a synchronized automaton then verifies the nested recurrence. We establish bounded discrepancy from the line of slope $c$, where $c^3-c^2+2c-1=0$, and show that the exact set of offsets from $\lfloor cn\rfloor$ is $\{-1,0,1,2\}$. We construct an explicit 26-letter non-erasing morphic presentation of the first-difference word, prove that its least balance constant is 4, and give an effective procedure for enclosing the global discrepancy extrema to arbitrary accuracy. We formalize the recurrence identification, six-decimal discrepancy bound, exact offset set, concrete morphic identity, least balance constant, and an effective extrema algorithm in Lean. Separate exact computations refine the numerical enclosures.

We determine generic completion-fiber dimensions for unit-norm tight frames when each partially observed column has exactly two missing coordinates. We work on the complex algebraic variety defined by the real frame equations. The missing coordinate pairs form a labelled multigraph. For every $d\ge4$ and every frame length $R\ge N_d:=\binom{d+1}{2}-1$, we prove that the missing-pair Jacobian matroid is the Rado matroid of an explicit subspace arrangement. Its rank is therefore given by a graph-theoretic minimum formula and is independent of $R$; the complementary rank defect is the generic fiber dimension. The main step is a realization theorem at length $N_d$. It combines a transversal–graphic matroid partition with a zero-coordinate moment submersion and a compatible decomposition of a generic residual matrix. Basis extension and stability under fully observed columns then yield the rank formula at every larger length. We also give an explicit defect formula in dimension four and a local real counterpart on the smooth real frame locus. A common-rotation obstruction bounds the smallest uniform stabilization threshold from below by $2d-1$.

We establish new bounds for incidences between a point set P and a family L of n-flats in PG(n+d,q). For fixed dimensions, our bound on the incidence discrepancy has an explicit piecewise-linear exponent in $\log_q|L|$, improving the classical estimate of Haemers and the Kong-Tamo bound in specified ranges of the number of flats. Matching constructions establish sharpness up to constant factors in several parameter ranges. The proof is combinatorial and avoids spectral and Fourier analytic methods. As applications, we obtain improved estimates for rich flats and exceptional orthogonal projections, together with stronger lower bounds for Furstenberg sets in certain ranges where the fraction of prescribed directions is small.

An outerplanar graph is maximal if no edge can be added without losing outerplanarity. Lin and Ning determined the outerplanar graph with the largest spectral radius, and the maximizer is a maximal outerplanar graph. We determine the minimizer. In this paper, we prove that every $n$-vertex maximal outerplanar graph $G$ satisfies $\rho(G)\ge\rho(F_n)$, where $F_n$ is the zig-zag triangulation of the $n$-gon, that is, the square of the path on $n$ vertices, with equality if and only if $G=F_n$. The proof uses three local operations on maximal outerplanar graphs, each of which strictly decreases the spectral radius: the first reverses the way a piece is attached along a chord, and the second and third move a piece from one vertex to its twin across a chord when the twin carries nothing or a single ear, respectively. A graph at which no operation applies is $F_n$, or has spectral radius greater than $4$, or consists of a central triangle with three zig-zag blades of at least three triangles each and has at most $15$ vertices; in the last case it contains one of two explicit graphs on $12$ vertices whose spectral radius exceeds that of $F_{15}$. Since $\rho(F_n)<4$ for all $n$, this completes the proof. The numerical inequalities used along the way are certified by explicit integer vectors with small entries.

We give a direct, self-contained construction of the three sporadic Fischer groups $\mathrm{Fi}_{24}'$, $\mathrm{Fi}_{23}$, and $\mathrm{Fi}_{22}$ from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions generate the full Fischer $3$-transposition group $\mathrm{Fi}_{24}$. Its commutator subgroup gives $\mathrm{Fi}_{24}'$, while $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ arise as centralizer quotients associated with one and two commuting transpositions. The root and frame geometry determines the group orders and leads to elementary proofs of simplicity, as well as natural rank-three actions and nonsplit central extensions. The construction uses standard facts about the Golay code, Parker's loop, and $M_{24}$. It does not use the Monster or previously known existence or order results for the Fischer groups. Fischer's classification and later recognition theorems are used only for the final identification.

We develop a discrete Morse-Bott theory for Lefschetz complexes with real-valued incidence functions and finitely many cells in each dimension. Our main result is a reduction procedure, based on four elementary operations called Moves, for computing the remainder series $R_t$ associated with the Morse-Bott inequality. These Moves preserve $R_t$, and the procedure recursively constructs, using finitely many Moves in each dimension, a disjoint union of two-cell elementary blocks with vanishing Poincaré series. The resulting decomposition computes the coefficients of $R_t$ by counting the blocks in the corresponding dimensions. The same procedure also computes the Betti numbers of the original complex by counting the cells removed as isolated cells. As consequences, we obtain the nonnegativity of the coefficients of $R_t$ and the Morse-Bott inequality for Lefschetz complexes. The method applies in particular to CW complexes, where the nonnegativity is obtained by identifying each coefficient of $R_t$ with the number of elementary blocks in the corresponding dimension.

We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$ bounds the description complexity, $p$ is the characteristic of $\mathbb{F}$, and $1/p=0$ in characteristic zero. We also prove this bound for incidences between points and distinct geometrically irreducible components of a two-parameter polynomial family, allowing singular and nonreduced members. The proof extends Lewko's interpolation and contact-multiplicity method from lines to algebraic families. Applications include rich components, polynomial values on difference sets, and polynomial expansion.

We prove that the five-column recursive-line Zarankiewicz number satisfies $z_{RL}(m,5)=3m+5$ for every integer $m\ge 12$. The proof is based on an explicit $12\times5$ seed configuration combined with a recursive one-row extension scheme. The seed attains the five-column cell bound and contains no unoccupied cells. Two selected pairs in the seed are opened and replaced by parallel paths, ensuring that each inserted row increases the total number of augmented edges by three while preserving the required simple configuration. The main challenge lies in verifying the strengthened recursive-line condition ${\rm (RW3+)}$ uniformly across all extension lengths. To this end, we develop a distance-reducing rectangle lemma that transfers certified inner-product relations along extension paths, reducing relations between distant labels to those at smaller path distances. This confines the verification for arbitrarily long extensions to a finite collection of seed and interface certificates. The resulting construction successfully satisfies pair identification, preserves distinct selected-edge classes, and certifies the orthogonality of distinct edge representatives. Furthermore, combining this constructed lower bound with the parameter hierarchy yields $z_2(m,5)=z_{RL}(m,5)=3m+5$ throughout this range.

For an integer $r\geq 3$ and real numbers $a_1,\ldots,a_r\in(0,1)$, let $a=a_1\cdots a_r$. We show that \[ \prod_{i=1}^r(a_i+a_i^2+\cdots+a_i^r)\geq \prod_{i=1}^r(a_i+a_i^2+\cdots+a_i^{r-1}+a). \] This inequality enables us to bound the measure of $r$-cross $t$-intersecting families.

Let ${S}_n(321)$ and ${S}_n(\underline{321})$ denote the sets of $n$-permutations avoiding the classical pattern $321$ and the consecutive pattern $\underline{321}$, respectively. Permutations are in bijection with Lehmer codes, a type of inversion sequence. Using Lehmer codes, we create the corresponding weighted posets ${L}_n(321)$ and ${L}_n(\underline{321})$, where the weight of a code is the inversion number of its permutation. We show that there are $2^{n-2}$ maximal elements of ${L}_n(321)$, while the maximal elements of ${L}_n(\underline{321})$ are enumerated by the Padovan numbers. We also show that when $n$ is even, each of these posets has a unique maximum weight element, and that when $n$ is odd, there are two maximum weight elements. These maximum weight Lehmer codes correspond to the pattern avoiding permutations with maximum inversion number.

Hyun, Kim, Wu and Yue constructed optimal and minimal binary linear codes from order ideals of hierarchical posets with two levels. Two different generalizations of the underlying antichain (simplicial complex) setting to odd characteristic are known: down-sets of $\mathbb{F}_p^n$ under the componentwise order, and support-closed subsets of $\mathbb{F}_q^m$. No generalization of the poset setting itself has appeared. We introduce generalized order ideals of a poset of order $p-1$, obtained by attaching multiplicities in ${0,\dots,p-1}$ to the elements of a poset, and study the two natural notions of order ideal that arise for hierarchical posets with two levels. Whenever the ideal meets the upper level, the resulting defining sets are neither down-sets nor support-closed. We determine the weight distributions of the associated complement codes, exhibit a family of Griesmer codes in which the upper element carries an arbitrary multiplicity, and, via the characteristic function of a generalized order ideal, obtain an infinite family of minimal $p$-ary codes of length $p^n-1$ and dimension $n+1$ violating the Ashikhmin–Barg condition.

We prove an $e$-positive formula for the chromatic symmetric function of proper circular arc graphs solving the $q=1$ case of Ellzey's conjecture. In doing so, we provide a new proof of the $e$-positivity of unit interval graphs, which alongside Guay-Paquet's reduction gives a new proof of the Stanley–Stembridge conjecture. We define color matrices, which count proper colorings, and tableau matrices, whose entries are nonnegative rational numbers and ratios of elementary symmetric functions. We prove the two matrices are related by a single family of change of basis matrices, which become invertible after restricting to finitely many colors, and we show the chromatic symmetric function of proper circular arc graphs comes from taking the trace of these matrices. Using Hikita's tableaux, this gives an explicit formula for the chromatic symmetric function of a proper circular arc graph as a weighted sum over tableaux whose first and last $k$ vertices lie in the same columns.

We prove that every finite loopless signed multigraph $\Sigma=(G,\sigma)$ satisfies $\chi'(\Sigma)\leq\Delta(G)+\mu(G)$, where $\chi'(\Sigma)$ is its chromatic index, and $\Delta(G)$ and $\mu(G)$ are the maximum degree and maximum multiplicity of $G$, respectively. This bound is sharp, even when both positive and negative edges are present, and generalizes both Vizing's theorem for ordinary multigraphs and Behr's theorem for signed simple graphs.

Let $p$ be an odd prime, $q=p^k$, and $r=p^s$ with $s\mid k$. We study $\mathbb{F}_r$-valued generalized Maiorana-McFarland functions $H(x,y,z)=P^{(z)}(x)+\operatorname{Tr}_s^k(yz)$ on $V_n\times\mathbb{F}_q\times\mathbb{F}_q$. The ingredients $P^{(z)}$ are vectorial dual-bent functions with weakly regular components, constant on the cosets of $\mathbb{F}_r^*$ in $\mathbb{F}_q^*$, and, for $z\ne 0$, of inversion type with respect to $t\mapsto t^{-1}$. Under a mild condition on $P^{(0)}$, we prove that the partition into the $\mathbb{F}_r$-valued level sets of $H$, refined according to the $\mathbb{F}_r$-lines of the relevant coordinates, is Fourier-reflexive. It therefore induces a translation association scheme with $\frac{q-1}{r-1}(r+1)+r$ classes (one fewer in a degenerate case), symmetric when the ingredients are even. For $s=1$ this recovers recent scalar constructions, while for $s=k$ it yields $(2q+1)$-class schemes from vectorial bent functions whose components may all be non-weakly regular. We also construct fusions along $\mathbb{F}_r$-subspaces, obtaining $(2r+1)$- and $(3r+2)$-class schemes and connections between different levels. Counterexamples show that the inversion, coset-constancy, subspace, and weak-regularity hypotheses cannot in general be omitted. To our knowledge, these are the first association schemes obtained from vectorial bent functions with non-weakly regular components.

Let $u=(u_n)_{n\ge0}$ be a binary sequence, and define \[ X_0=1,\qquad X_{n+1}=2X_n+u_n \qquad(n\ge0). \] Let $A_u$ be the set of all nonempty finite subset sums of the sequence $(X_n)$, and let $L_u(N)$ denote the maximum length of an arithmetic progression contained in $A_u\cap[1,N]$. We prove that there are absolute constants $c>0$ and $N_0\geq 1$ such that, for every binary sequence $u$, \[ L_u(N)\ge \exp\!\left(c\sqrt{\frac{\log N}{\log\log N}}\right) \] for all $N\geq N_0$. Moreover, if the random variables $u_n$ are independent and uniformly distributed on $\{0,1\}$, then, almost surely, \[ L_u(N)\ll_u N^{2/3}\exp\!\left(C\sqrt{\log N\log\log N}\right), \] where $C>0$ is an absolute constant. Furthermore, every eventually periodic binary sequence satisfies $L_u(N)\gg_u N^{1/2}$.

The Berger-Charpin conjecture predicts that the permutation automorphism group of a cyclic code is generally the $q$-affine group, which is generated by the shift and the Frobenius multiplier on the index set. The word generally is not made precise in the literature, and the goal of this series of papers is to study when the Berger-Charpin conjecture or its variants hold. In this first part, we establish a natural connection between the theory of cyclic codes and the theory of cyclotomic association schemes (equivalently, Schur rings over cyclic groups). Using it, we show that the $q$-affine group is not the correct group to expect for certain code lengths. We characterize these lengths by an arithmetic condition, and show that they have density zero.

A graph G is uniquely C_t-saturated if G contains no cycle of length t and, for every edge e of the complement, G+e contains exactly one cycle of length t; it is nontrivial if it has at least t vertices. Wenger and West proved that no nontrivial uniquely C_6- or C_7-saturated graphs exist, and conjectured the same for every t >= 6; the case t=8 was verified but never published, and t >= 9 has remained open. We prove the conjecture for t=9: there is no nontrivial uniquely C_9-saturated graph. The proof is a case analysis on the length L of a longest even cycle of length at most 12 (L is 4, 6, 8, 10, or 12). The cases L=12, L=10, and L=4 are settled by hand; for L=6 and L=8, hand classifications of the components outside the cycle reduce each case to a bounded finite family of configurations, eliminated by a short, replayable computer enumeration with known-answer controls and completeness certified by zero cap hits. Along the way we prove, in sharpened form, the t=9 instance of a structural lemma Wenger and West stated without proof.

In this note we prove $\mathrm{forb}(n,\mathbf{1}_{r-2}\times C_4)=\Theta(n^{r-1/2})$, thus providing a series of counterexamples of a 2005 conjecture of the first two authors. Here $\mathbf{1}_{r-2}\times C_4$ is the $r$-extension of the 4-cycle, that is the $r$-uniform hypergraph of four edges obtained by joining a set of $r-2$ vertices to each edge of $C_4$. $\mathrm{forb}(n,F)$ is the largest number of edges of a simple hypergraph $G$ on $n$ vertices without trace (induced subhypergraph) $F$.

A well known result due to Lakshmibai and Sandhya states that smooth Schubert varieties of type A correspond to permutations that avoid the patterns 3412 and 4231. It was later shown by the second author and Slofstra that staircase diagrams over a simply-laced Dynkin diagram are in bijection with smooth Schubert varieties of the corresponding type. We explore how the poset structure of staircase diagrams is connected with pattern avoidance on smooth permutations. As an application, we enumerate several subclasses of smooth permutations which are characterized by pattern avoidance. These subclasses include the class of polished permutations which were studied by Gaetz and Gao.

We report a novel polyhedral surface of genus~3 embedded in $\mathbb{R}^3$ with eight planar, simple, non-convex nonagonal faces, 24 vertices and 36 edges, in which every two faces share at least one edge: 20 pairs of faces share one edge and 8 pairs share two collinear edges. Its face planes are $3x-4y-2z=5$, $-2x+5y-5z=3$ and their images under the half-turns about the three coordinate axes, and all vertices are rational. The polyhedron has the same face vector, face sizes and number of edge multiplicities as the polyhedron described by Mizhaev, but it is not combinatorially equivalent to it. Our realisation has the symmetry group $D_2$ of order~4, whereas Mizhaev's polyhedron has a rotoreflection symmetry. The example was found by a computational geometric search, and all its properties were verified in exact rational arithmetic.

Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, the derangement graph $\Gamma_G$ is the graph whose vertex set is $G$, and two vertices $g$ and $h$ are adjacent if the ratio $h^{-1}g$ is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement graph of the transitive permutation group corresponding to the natural action of $\operatorname{PGL}_{2}(q)$ on the projective line $\operatorname{PG}_{1}(q)$ is \begin{align*} \operatorname{Aut}(\Gamma_{\operatorname{PGL}_{2}({q})}) = \left(L_{\operatorname{PGL}_2(q)}\times R_{\operatorname{PGL}_2(q)}\right) \rtimes \left(\langle \psi \rangle \times \gamma_{\operatorname{Aut}(\mathbb{F}_q)}\right), \end{align*} where $L_{\operatorname{PGL}_2(q)}$ is the left-regular representation of $\operatorname{PGL}_2(q)$, $R_{\operatorname{PGL}_2(q)}$ is the right-regular representation of $\operatorname{PGL}_2(q)$, $\gamma_{\operatorname{Aut}(\mathbb{F}_q)}$ is the group of conjugation by elements of $\operatorname{Aut}(\mathbb{F}_q)$, and $\psi: \operatorname{PGL}_2(q) \to \operatorname{PGL}_2(q)$ such that $\psi(x) = x^{-1}$.

Santos asked when intersections of dilated Hadamard simplices with cubes are integral, smooth, or normal, in a prescribed affine lattice. We construct a nonintegral example in dimension eleven and prove that no smaller-dimensional example exists. We characterize smoothness completely and show that every smooth member of this family is normal. An explicit example in dimension fifteen shows that integrality alone does not imply normality. For Sylvester simplices of order at least sixteen, we establish a sharp uniform integrality bound and construct counterexamples immediately below it. We also obtain sufficient normality bounds for general Hadamard simplices and stronger bounds for the Sylvester family. The proofs use integer decomposition for boxes with separated corner cuts and rounding under three signed slab constraints. All numbered results have formal counterparts verified in Lean.

We prove that there is an absolute positive constant $c$ such that every bipartite graph with $N$ vertices in each part and at least $N^2/2$ edges contains a complete bipartite graph $K_{t,t}$ whenever $N\ge c\, 2^{t}$. This improves the classical Kővári-Sós-Turán bound requiring $N$ of order $t\,2^t $. As a consequence, the diagonal bipartite Ramsey number has upper bound $b(t,t)=O(2^t)$, improving the previous best bound $b(t,t) = O(2^t\, \log t )$ due to Conlon. The proof was found by GPT-6 Astra, and the method will probably have further applications.

The following is a natural and fundamental question for a graph $G$: if an induced subgraph $H$ of $G$ has $x$ more vertices than a maximum independent set, what can be said about the maximum degree of $H$ as a function of $x$? The case $x=1$ is already of considerable interest. For example, in his celebrated proof of the sensitivity conjecture, Hao Huang showed that every induced subgraph of the hypercube $Q_n$ on more than $2^{n-1}$ vertices has maximum degree at least $\sqrt{n}$. Chung, Fúredi, Graham, and Seymour proved that this bound is tight. In this paper, we study this question when $G$ is either the $n$-fold Hamming product or the $n$-fold tensor product of a triangle. For both graphs, we determine the exact minimum possible average degree of an induced subgraph of a prescribed size. We also prove that the tensor product exhibits several Huang-like phenomena. For the Hamming product, we show that, for several size densities and large $n$, the minimum possible maximum degree is asymptotically equal to the minimum possible average degree. Finally, we extend several of the results from the triangle to an arbitrary complete graph $K_k$.

Let $G_n$ be the bipartite complement of the even cycle $C_{2n}$, and let $J_{G_n}$ be its binomial edge ideal. For $n\geq5$, we study the interaction between the combinatorial structure of $G_n$ and the algebraic invariants of $S_{G_n}/J_{G_n}$. Our main combinatorial ingredient is a classification of the subsets of $V(G_n)$ having the cut point property, obtained through an analysis of disconnected induced subgraphs of $G_n$. We use this classification to describe the minimal primes and to study dimension and degree-theoretic invariants of $J_{G_n}$ and $S_{G_n}/J_{G_n}$. We also study local Vasconcelos numbers, graded Betti numbers and the Hilbert series, and obtain information on projective dimension and depth. Finally, we analyze induced paths in $G_n$ and derive corresponding bounds for the Castelnuovo–Mumford regularity.