Papers

Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write ${w}(P)$ for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound $\operatorname{gap}(P)\le 2 {w}(P)-1$. Second, for every $L>0$ we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least $L$, with $0$ and $|P|+1$ added as endpoints. Finally, for every $r\in\mathbb{N}$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.

For $1\le s<t$ and any graph $G$, the weakened Gallai-Ramsey number $gr^t_s(G)$ is defined to be the least $p\in \mathbb{N}$ such that every Gallai $t$-coloring of the edges of $K_p$ (i.e., a $t$-coloring that lacks rainbow triangles) contains a subgraph isomorphic to $G$ whose edges use at most $s$ of the colors. In the case of a book graph $B_n:=K_2+nK_1$, Jakhar and Moun determined the values $gr^3_2(B_3)=6$ and $gr^3_2(B_4)=7$. In this paper, we extend their results to $t>3$ colors, and we determine the values of $gr^3_2(B_n)$ for $5\le n\le 15$. General lower bounds for $gr^t_2(B_n)$ are also given.

In even characteristic, the first subconstituent of an orthogonal graph at a singular point is shown to be the complement of an affine polar graph. This identification yields the full automorphism group as an affine semisimilarity group and provides explicit extensions to the ambient graph. Maximum independent sets are affine cosets of maximal totally singular subspaces, giving the independence and chromatic numbers. In plus type, Delsarte cliques correspond bijectively to ovoids through the base point, leading to a sharp rank threshold. Over $\mathbb{F}_2$, we determine the exact maximum clique number in every dimension, prove that all maximum cliques form a single orbit, and obtain the core of every first subconstituent. A uniform Witt-extension argument gives a short proof of triple transitivity for the binary local graphs.

Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics of the expected number of adjacent twin pairs in every dimension. In dimension two, more generally, for each origin-symmetric convex connection body $K\subset(-1/2,1/2)^2$, the twin count converges to a Poisson variable with mean $\operatorname{area}(K^\circ)/16$; hence the zero-one law fails for all fixed $0<r<1/2$ in the $L^\infty$ and Euclidean models. At each critical component threshold $n^k r_n^{D(k-1)}\to a\in(0,\infty)$, the numbers of components of the feasible connected $k$-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For $D\ge3$, we also construct a definable common-neighborhood configuration of probability order $1/n$.

We construct an explicit real quantum Latin square of order six with cardinality $29$, the last unresolved value in the order-six spectrum. We first construct a punctured $6 \times 6$ array in $\mathbb{R}^5$ whose punctured rows and columns are orthonormal bases, and then adjoin a common diagonal vector in an orthogonal one-dimensional summand. The six diagonal entries lie on one ray. Of the thirty off-diagonal entries, two rays occur twice and the other twenty-six occur once; exact support and coordinate-ratio comparisons establish this count. Together with the known constructions, this closes the cardinality spectrum of quantum Latin squares of order $6$.

For $X\in\{\mathbb{N},\mathbb{Z}\}$, let $\alpha_X(\ell)$ and $\beta_X(\ell)$ denote the supremal upper and lower densities of subsets of $X$ admitting $\omega$-permutations without monotone $\ell$-term arithmetic progressions. We strengthen the published lower bounds for the three-term upper-density parameters by proving \[ \alpha_{\mathbb{N}}(3)\geq\frac23,\qquad \alpha_{\mathbb{Z}}(3)\geq\frac23. \] We also prove $\beta_{\mathbb{Z}}(4)=1$ by constructing four-permutable subsets of the integers whose lower symmetric densities approach one.

We study (binary) nearly-perfect covering codes, which are codes that attain the Van Wee bound with equality. They act as the covering counterparts to nearly-perfect error-correcting codes, which attain the Johnson bound with equality. These codes have been completely classified for covering radius $R=1$. We prove that no code with $R\geq 2$ can attain the original Van Wee bound with equality, since it omits the dependence on the minimum distance of the code. We refine the bound to account for the minimum distance and show some nearly-perfect covering codes. By proving some structural properties of such codes, we prove all nearly-perfect covering codes with $R=2,3$ must be equivalent to the codes we showed. We also prove that for any $R\geq 3$, there are at most a finite number of nearly-perfect covering codes.

We construct a 321 point set in R^63 that cannot be partitioned into 64 subsets of smaller diameter, proving b(63)>=65. Starting from the G_2(4) Euclidean representation and the Jenrich Brouwer 320 point core in dimension 63, we add one projected and rescaled point while preserving the relevant clique obstruction. The example and proof were generated entirely by ChatGPT using GPT 5.6 Sol. The author has personally verified the result and assumes responsibility for that verification, but claims no credit for the originality of the construction.

Let $S$ be a numerical semigroup with multiplicity $\mult$, conductor $\cc$, embedding dimension $\ee$, type $\typ$ and genus $\gnus$, and let $\nn=\cc-\gnus$. Wilf's conjecture asserts that $\ee\,\nn\ge\cc$; the inequality $\gnus\le\typ\,\nn$ of Fröberg, Gottlieb and Häggkvist settles it when $\typ\le\ee-1$. The Apéry set of $S$ with respect to any $s\in S\setminus\{0\}$ carries a partial order whose maximal elements are the pseudo-Frobenius numbers translated by $s$; for $s=\mult$ its minimal elements are the minimal generators other than $\mult$. Comparing the two extremal statistics bounds the type by $\typ\le\ee-1+\Xii(S)\le\ee-1+\Theta(S)$, where $\Theta(S)$ measures the redundancy of the covering of the gaps of $S$ by the pseudo-Frobenius numbers and $\Xii(S)$ refines it. With an exact decomposition of the Wilf number this yields the genus bound $\gnus\le\ee-1+\typ(\nn-1)$, strictly stronger than $\gnus\le\typ\,\nn$ precisely when $\typ\ge\ee$, and reduces Wilf's conjecture to an inequality free of $\cc$ and $\nn$. We determine the equality case of $\gnus\le\typ\,\nn$, recovering a classification of Singhal; answer a question of Moscariello and Sammartano whenever $\ee\ge\typ+1$; and correct Kaplan's classification of the equality case for $\cc\le2\mult$, from which an infinite family is missing.

Let \(k\geq 3\), \(n\geq 1\), and let \(H_{k,n}=\Sh(C_k,n)\) be the group generated by the standard \(k\) pile perfect shuffle and cyclic pile permutation on a deck of \(kn\) cards. We prove that \(H_{k,n}\) is \(2\)-transitive whenever \(n\) is not a power of \(k\). Residual commutators give translations supported on two pile labels, and a strongly connected digit digraph propagates these translations throughout the deck, a separate argument resolves the antipodal support case. We then combine this result with fixed point ratio bounds for primitive groups and explicit boundary calculations to determine \(H_{k,n}\) for all \(k\) and \(n\). If \(n=k^f\), then \(H_{k,n}\cong C_k\wr C_{f+1}\). If \(k=4\) and \(n=2\cdot4^j\), then \(H_{4,n}\cong\AGL(2j+3,2)\). In every other case, \(H_{k,n}\) is \(\Alt(kn)\) or \(\Sym(kn)\), according to the parity of its generators. This proves Conjecture~1.10 of Amarra, Morgan and Praeger and Conjecture~5.1 of Xia, Zhang, Zhang and Zhu. More generally, we classify \(\Sh(P,n)\) for every pile group \(P\) containing \(C_k\), and obtain the odd \(k\) part of their Conjecture~5.2.

A coloring of the vertices of a hypergraph is called balanced if the sizes of the color classes differ by at most one. We say that a hyperedge is rainbow if its elements have pairwise distinct colors. In this paper, we provide a general upper bound on the balanced upper chromatic number of arbitrary linear hypergraphs, that is, the largest integer $k$ such that there exists a balanced $k$-coloring of the vertices of the hypergraph without rainbow hyperedges. We focus on the cube $C_t^n$, defined as the linear hypergraph whose vertices are the lattice points in $[0,t-1]^n$, and whose hyperedges are the sets of $t$ collinear points. We determine the exact balanced upper chromatic number of $C_t^n$ for $t\geq 4n-2$. For smaller values of $t$, we present bounds and determine this parameter (with few exceptions) in dimensions $2$ and $3$.

We study descendant integrals on moduli stacks of simplicially stable curves. For each finite collision complex, we factor the reduction morphism into elementary wall crossings and obtain a reconstruction formula in terms of ordinary Witten-Kontsevich correlators. We use the formula to derive corrected Virasoro recursions. We then package all finite-support theories in a completed disjoint-support Fock space, where the extended potential is a cumulant translate of the Witten-Kontsevich potential.

A vertex cover of a graph $G$ is a subset of vertices $C \subseteq V(G)$ such that every edge of $G$ is incident to at least one vertex in $C$. The vertex cover number of $G$ is the minimum cardinality of a vertex cover of $G$ and is denoted by $\tau(G)$. A snark is a connected, bridgeless, cubic graph that has an edge chromatic number of four, meaning its edges cannot be properly colored with only three colors. In this work, we investigate the problem of determining the value of a minimum vertex cover for classes of snark graphs. Given a positive integer $k$, we firstly prove that determining whether an arbitrary snark has a vertex cover $C$ with size $|C| \leq k$ is an NP-complete problem. Secondly, we determine the vertex cover number $\tau(G)$ for several subclasses of snark graphs, such as Flower snarks, Goldberg snarks, Generalized Blanuša snarks and Loupekine snarks.

Let $v_1,\ldots,v_T\in B_2^m$ be fixed in advance and revealed sequentially, and assume that $\|v_t\|_\infty\leqslant d^{-1/2}$ for some $d\geqslant 1$ and every $1\leqslant t\leqslant T$. There are absolute constants $L,C,c>0$ and a randomized online signing such that $$\mathbb{P}\left\{\max_{k\leqslant T}\left\|\sum_{t=1}^k\varepsilon_t v_t\right\|_\infty>6L\right\} \leqslant CT\exp\left(-\frac{cd}{\ln^2(ed)}\right).$$ Consequently, constant prefix discrepancy holds with probability at least $1-\varepsilon$ once $d$ is at least $C\ln\frac{3T}{\varepsilon}\left[\ln\left(e+\ln\frac{3T}{\varepsilon}\right)\right]^2$. In particular, every fixed sequence of vectors $a_t\in[-1,1]^m$ with at most $d$ nonzero coordinates admits an online signing with prefix discrepancy $O(\sqrt d)$ and failure probability at most $CT\exp[-cd/\ln^2(ed)]$. We also prove a nonuniform version in which the failure probability depends on the individual parameters $d_t=\|v_t\|_\infty^{-2}$, and a lower bound showing that a universal constant prefix discrepancy is impossible when $d=o(\ln T)$. We identify the corresponding $\ln^2 d$ barrier for the compact-potential method and extend the argument to general symmetric target bodies admitting a quadratic smoothness estimate.

The main result of this article shows that the quiver mutation class of a malleable real Morsification uniquely determines the integral monodromy module of a plane curve singularity. In particular, we show that the quiver mutation class of a malleable divide determines the complex topological type of an irreducible plane curve singularity. This establishes the algebraic-to-topological implication of a conjecture of S.~Fomin, P.~Pylyavskyy, E.~Shustin and D.~Thurston in the malleable irreducible case. The result is proven by developing representation-theoretic techniques based on an equivariant Euler pairing in the derived category of continuous finite-dimensional dg modules over a differential bigraded Ginzburg algebra. A key step uses these techniques to show that the quiver mutation class of a plabic fence uniquely recovers the torsion part of the Alexander module of the associated smooth link.

A classical problem in graph theory known as the Italian domination number(also called Roman 2-domination number), involves assigning labels of 0, 1,or 2 to each node v. The goal is to ensure that every node with a label of 0 has a sum of labels of the nodes in its closed neighborhood that is 2 or greater. In computer systems, it is coined encompassing a robust cyber security strategy that will protect networks from potential threats, such as hacking, malware, and unauthorized access, by deploying security measures to provide the highest level of protection while reducing the misuse of resources. Toeplitz graphs are a special kind of graphs built over Toeplitz matrices from linear algebra, which are matrices with constant straight diagonal members. In this paper, we provide a detailed analysis regarding the Italian domination numbers for every Toeplitz graph family. We provide comprehensive results on Italian domination numbers across multiple graph families and identify the specific values at which the Italian domination number alters with increasing generator values.

In phylogenetics, Metropolis-Hastings methods are commonly used to sample phylogenetic trees or networks, for example from Bayesian posteriors. These methods generally use transitions that distinguish all nodes involved, and thus require fully labelled representations of phylogenetic networks. We argue that sampling leaf-labelled phylogenetic networks demands a correction for the number of fully labelled representatives of a leaf-labelled network, or, equivalently, for its internal symmetry. Without correction, there is a danger of undersampling networks with internal symmetries. We show that this correction can be realized by a quotient construction on the Metropolis-Hastings Markov chain, which, in practice, requires the calculation of the size of the network's automorphism group. Using $\mu$-vectors, we show that the automorphism group is trivial for orchard networks, and thus also for tree-child networks and trees. This implies that a correction for symmetry is not needed when sampling only from such network classes. More generally, using our Python implementation of the algorithms in this paper, we show that using $\mu$-vectors can significantly speed up calculations of automorphism group sizes and thus of Metropolis-Hastings sampling of leaf-labelled networks.

The degree/diameter problem asks for the largest order of a graph with a given diameter and maximum vertex degree. This has been widely studied and given rise to a recent variation for mixed graphs (graphs with both undirected edges and directed arcs), where an additional bound is placed on the maximum directed out-degree of any vertex. Both problems have applications to network design. Counting the possible number of vertices at each distance from a given vertex gives a bound on the order of a mixed graph satisfying the degree and diameter constraints (the mixed Moore bound). In this paper, we settle an open problem posed by Tuite and Erskine concerning the total regularity of mixed graphs whose order is one less than the mixed Moore bound (almost mixed Moore graphs). We use this result to show that the three known almost mixed Moore graphs of diameter at least three are the only such mixed graphs.

An analog of the classical Kronecker-Weyl theorem for weak uniform distribution is obtained for an arbitrary countably infinite independent subset of a discrete abelian group using a type of interpolation sets called epsilon-Kronecker sets. Specifically, the topological size of the set of homomorphisms inducing weakly uniformly distributed sequences is established by showing that it forms a dense G-delta set.

In this work, we present novel approaches for constructing linear codes over $\ZZ_4$ from the known ones. We succeeded in obtaining new linear codes, many of which are optimal. In particular, we found all optimal codes for $k_1=2,~k_2=0$ and many optimal codes for $k_1=3,~k_2=0.$

Let $X\subset\mathbb{R}$ be finite and let $\gamma(t)=(t,f(t))$, where $f$ is strictly convex. We show that \[ J_3(\gamma(X)) =\#\{(x_1,\ldots,x_6)\in X^6:\sum_{i=1}^3\gamma(x_i)=\sum_{i=4}^6\gamma(x_i)\} \ll_{\epsilon}|X|^{3+\epsilon}. \] When specialized to the parabola, our result implies near-optimal estimates for the number of solutions to the diameter-free quadratic Vinogradov system. As a second application, we settle a conjecture from Krishnapur-Kurlberg-Wigman and Bombieri-Bourgain concerning lattice points on dilates of the unit circle. As a third application, we prove that $|A-A|\gg_\epsilon|A|^{5/3-\epsilon}$ and $|A+A|\gg_\epsilon|A|^{8/5-\epsilon}$ for any finite convex sequence $A\subset \mathbb{R}$.

Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved by a new multiplicative Fourier argument. The index-two residue identity yields a character-moment relation, and the odd characters with vanishing first moment form an exceptional spectrum of size at most $157\varphi(n)/1440<\varphi(n)/9$. A finite-group uncertainty principle then forces the four-term multiset to be invariant under negation, contradicting minimality. Apart from standard facts about primitive Dirichlet $L$-functions, the remaining argument is finite and requires neither asymptotic estimates nor computational verification.

We focus on the finite basis problem for flat semirings of the form $S(W)$, where $W$ is an arbitrary set of nonempty words. We prove that $S(W)$ generates a Cross variety (and hence is finitely based) whenever every word in $W$ has length at most $3$, whereas it is nonfinitely based whenever there exists $k \geq 3$ such that $W$ is $x^{k+2}$-free but not $x^{k+1}$-free. In particular, if $W_k$ denotes the set of all words of length $k$, then $S(W_k)$ is finitely based if and only if $k \leq 3$. Moreover, $S(W)$ is nonfinitely based whenever $W$ is finite and not $x^4$-free. These results provide a partial answer to an open problem raised by Jackson et al.~(J Algebra 611: 211–245, 2022).

We prove the conjecture of Huh, Matherne, Mészáros, and St.~Dizier that the normalization of every skew Schur polynomial in finitely many variables is Lorentzian. We first realize every nonzero skew Schur polynomial in finitely many variables as a specialization of a Schubert polynomial and prove that it is dually Lorentzian. The dual Jacobi–Trudi identity then identifies its normalization with the finite dual of a skew Schur polynomial obtained by rectangular complementation. As a consequence, skew Kostka numbers satisfy log-concavity inequalities along the root directions.

The $n$-dimensional hypercube $Q_n$ is the graph with vertex set $\{0,1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. For a graph $H$, let $\operatorname{ex}(Q_n,H)$ be the maximum number of edges in an $H$-free subgraph of $Q_n$. The hypercube Turán density of $H$ is defined by $\pi_{\square}(H)=\lim_{n\rightarrow\infty}\operatorname{ex}(Q_n,H)/|E(Q_n)|$. In this note, we prove \[ \frac{1}{8} \leq \pi_{\square}(C_{10}) \leq 0.36577. \] For the upper bound, we prove $\pi_{\square}(C_{10}) \leq \pi_{\square}(C_6)$, which, together with a result of Baber, gives the stated upper bound. For the lower bound, we prove that $\operatorname{ex}(Q_n,C_{10}) > |E(Q_n)|/8$ for every $n \geq 2$.

The Hajnal–Szemerédi theorem states that every graph with $\Delta(G)\le k-1$ admits an equitable $k$-coloring. Using a reduction implicit in Aboulker, Oijid, Petit, Rocton, and Simon (which itself builds on the literature surrounding feedback arc sets), we prove that this theorem is equivalent to each of four digraph analogues guaranteeing equitable acyclic $k$-colorings under various degree conditions. A minor modification of the proof method in makes the construction algorithmic, so the equitable acyclic coloring guaranteed by these statements can be found in polynomial time. The same reduction shows that the Ore-type Hajnal–Szemerédi theorem is equivalent to its four digraph analogues. These equivalences settle the conjecture of Treglown that every $n$-vertex digraph with $r\mid n$ and $\delta^{\max}(D)\ge(1-\frac{1}{r})n$ contains a $T_r$-factor, answer a question of Molla and Treglown by determining the exact Ore-type threshold for $T_r$-factors, and recover the directed Hajnal–Szemerédi theorems of Czygrinow, DeBiasio, Kierstead, and Molla as immediate corollaries of this reduction. We also discuss related problems for oriented graphs which do not follow immediately from the reduction mentioned above.

Suppose that ${R}^{\alpha}$ is a rhombus with side length $1$ and with an interior angle $\alpha$, where $0<\alpha\leq \frac{\pi}{2}$. Let $\triangle$ be an equilateral triangle with a side parallel to a side of ${R}^{\alpha}$ and let $\{\triangle_{n}\}$ be a collection of homothetic copies of $\triangle$. In this paper, we show the following two results: if $0<\alpha\leq\frac{\pi}{3}$ and the sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $\frac{\sqrt{3}}{4}(1+\cos\alpha+\frac{\sqrt{3}}{3}\sin\alpha)^{2}$, then these equilateral triangles can parallel cover the rhombus ${R}^{\alpha}$; if $\frac{\pi}{3}<\alpha\leq\frac{\pi}{2}$ and the sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $\frac{\sqrt{3}}{4}(1+\frac{2\sqrt{3}}{3}\sin\alpha)^{2}$, then they can parallel cover the rhombus ${R}^{\alpha}$. Furthermore, these bounds are optimal on their respective intervals.

Illingworth and Wood recently proposed the Dominating Hadwiger's Conjecture, a strengthening of Hadwiger's Conjecture which asserts that every graph with no dominating $K_t$-model is $(t-1)$-colorable. We prove two relaxations of this conjecture. First, we show that every graph with average degree $Ct (\log t)^2$ contains a dominating $K_t$-model for some absolute constant $C$. This bound improves on the $2^{t-2}$ due to Illingworth and Wood and is within an $O(\log t)$ factor from optimal. Second, we prove that the vertices of every graph with no dominating $K_t$-model can be partitioned into $t-1$ parts such that the subgraph induced by each part has bounded maximum degree.

In a proper vertex coloring $c$ of a graph $G$, a vertex $u$ is called a b-vertex if $u$ is adjacent to a vertex in every other color class. A ${\rm b}^{\ast}$-coloring is a proper coloring in which a b-vertex is adjacent to a b-vertex in every other color class. A Grundy coloring is a proper coloring obtained by the First-Fit (greedy) coloring procedure. A $z$-coloring of $G$ is a ${\rm b}^{\ast}$-coloring that is also a Grundy coloring. The ${\rm b}^{\ast}$-chromatic number (resp., $z$-chromatic number), denoted by ${\rm b}^{\ast}(G)$ (resp., $z(G)$), is the maximum number of colors used in a ${\rm b}^{\ast}$-coloring (resp., $z$-coloring) of $G$. Every graph admits a ${\rm b}^{\ast}$-coloring and a $z$-coloring that can be found using a polynomial-time coloring heuristic. Let ${\rm m}^{\ast}(G)$ be the largest integer $k$ such that a vertex of degree at least $k$ in $G$ has $k$ neighbors of degree at least $k$. We employ list-coloring techniques to prove that if $G$ has a girth of at least $7$, then ${\rm b}^{\ast}(G) = {\rm m}^{\ast}(G)+ 1$. A similar result is obtained for graphs of girth at least $6$ when ${\rm m}^{\ast}=3$. Finally, we obtain some results for the $z$-chromatic number. We prove that if the girth is at least $2m^{\ast}(G)+4$ and $G$ contains a specific tree as an ordinary subgraph, then $z(G)= m^{\ast}(G)+1$.

We construct a family of layer barriers for colour-biased tight Hamilton cycles in uniform hypergraphs. For every $k\ge 3$ and every $a\in\{0,\ldots,k-1\}$, we give a red–blue coloured $k$-graph that contains a tight Hamilton cycle, while every tight Hamilton cycle in the construction is perfectly colour-balanced. The construction underlying the higher-uniformity threshold conjectured by Behague, Clemen, Hyde and Morrison corresponds to the boundary case $a=0$ of this family. We show that interior choices of $a$ can yield strictly denser barriers. In particular, for $k=17$ and $a=8$, the asymptotic relative minimum vertex degree of our construction is \[ \frac{5761}{8192}\approx 0.703247, \] which exceeds the conjectured value $d_{17}\approx 0.699277$. This provides a counterexample to the proposed higher-uniformity threshold in Conjecture~6.1 of Behague, Clemen, Hyde and Morrison. Moreover, by choosing the layer appropriately as $k\to\infty$, the family contains barriers whose asymptotic relative minimum vertex degree is \[ 1-O\bigl(k^{-1/2}\bigr). \] Thus the interior members of the layer-barrier family exhibit substantially different behaviour from the previously considered boundary construction in large uniformity.