Papers

For positive integers $t$ and $q$ let $b_q(t)$ be the smallest integer $n$ so that any coloring of the edges of the complete bipartite graph $K_{n,n}$ with $q$ colors yields a monochromatic copy of $K_{t,t}$. We give an independent proof that $b_2(t)\le 128\cdot 2^t$ for every positive integer $t$. More generally, for $0< p\le 1/2$, every bipartite graph of edge density at least $p$, with both classes of size at least $128\cdot p^{-t}$, contains $K_{t,t}$. This gives $b_q(t)\le 128\cdot q^t$ for every integer $q\ge 2$ and a uniform consequence for Zarankiewicz numbers.

The 2-switch-degree $\text{deg}(G)$ of a graph $G$ is the number of 2-switches that can be performed on $G$; equivalently, it is the degree of $G$ as a vertex of the realization graph $\mathcal{G}(d)$ of its degree sequence $d$. We classify the prime graphs of 2-switch-degree at most 4, where a graph is prime if it is indecomposable with respect to the Tyshkevich composition and every vertex takes part in some 2-switch. From this classification we derive a sharp dichotomy that recovers the global shape of $\mathcal{G}(d)$ from a single one of its local degrees: if $d$ has a realization $X$ with $\text{deg}(X)=k\le3$, then $\mathcal{G}(d)$ is vertex-transitive and $k$-regular if and only if neither $T_{221}$ nor $\overline{T_{221}}$ is an induced subgraph of $X$. Moreover, for every $k\geq 4$ some prime graph of degree $k$ carries a 2-switch raising its degree to $2k-2$. As a further consequence, up to isomorphism there are only 10 realization graphs of prime graphs with degree at most 4.

We develop a submodular framework for the von Neumann entropy of discrete quantum convolutions, providing a noncommutative counterpart to the direct side of entropic additive combinatorics. We first introduce globally weighted quantum convolutions, which form compatible families indexed by admissible subsets of a fixed collection of inputs. Our main theorem reveals a polymatroidal geometry underlying their entropy growth: relative to any fixed admissible input block, the entropy gains admit a normalized, monotone, submodular extension to all subsets of the remaining inputs. The theorem yields convolutional strong subadditivity, quantum Ruzsa triangle inequality, and quantum entropic Plünnecke–Ruzsa inequalities for arbitrary input states. For repeated inputs, it gives sharp comparisons of entropy growth across admissible scales; in particular, the quantum doubling constant controls all higher admissible convolution entropies with optimal exponents. Together, these results bring submodular methods from additive combinatorics into the quantum setting and provide a systematic route to broad families of convolutional entropy inequalities.

Let \(N_t\) denote the least number of nodes in a positive cubature formula of degree \(t\) on \(S^2\). We prove that a formula of degree \(2m+1\) cannot have exactly \((m+1)(m+2)+1\) nodes for any \(m\ge2\). Excluding equality in the Fisher bound then gives \[ N_{2m+1}\ge (m+1)(m+2)+2 \] for \(m\ge3\), and known constructions yield the exact values \(N_7=22\) and \(N_9=32\). For general odd degrees, positive circle measures on Lobatto latitudes give an upper bound with quadratic coefficient \(13/8\), parity-dependent linear terms, and an \(O(m^{2/3})\) remainder. We determine the sharp constant \(49\sqrt[3]{3}/72\) for the scalar remainder in this construction. Lower bounds are obtained from continuous weighted LP–Turán inequalities and radial caps whose Helmholtz companions are nonnegative measures. We prove that the cap functional admits a maximizer at each fixed admissible support radius and derive explicit finite-degree lower bounds by a positivity-preserving transfer to the sphere.

We show that compact locally symmetric manifolds $M$ with universal cover the symmetric space $X$ for $SL(n,\mathbb{R})$ form a topological higher $d$-expander family for $d\leq n/8$. We prove the same statement for $SL(n,\mathbb{R})$ replaced by a split simple non-compact real Lie group $G$ and for $d$ linear in the rank of $G$. We accomplish this by showing that minimal submanifolds of low codimension in such $M$ must have volume comparable to the volume of $M$. Our proof is based on a new monotonicity formula for minimal submanifolds of $X$, together with bounds on the decay of matrix coefficients for unitary representations of higher rank Lie groups. We also give the first locally symmetric example of power-law systolic freedom. This paper partially supersedes \cite{fl24}.

A well-known problem of Erdős and Hajnal from the 1960s asks whether every graph with huge chromatic number contains a subgraph with large girth and large chromatic number. Very recently, Kohlmeyer and Kruer provided a strong negative solution to this problem: a construction of triangle-free graphs with arbitrarily large chromatic number whose subgraphs with no four-cycle have chromatic number at most $6$. The purpose of this exposition is to explain the construction method, relate it to relevant literature, and optimise the bound '$6$' to '$3$'.

We present a "myopic" Tutte polynomial for graphs on $\mathbb{R}P^2$ which takes only nullhomologous spanning subgraphs as input. It recovers the generalized Krushkal polynomial and Drobotukhina's analogue of the Jones polynomial for alternating, nullhomologous links in $\mathbb{R}P^3$. We use this myopic Tutte polynomial to prove an analogue of the Kauffman-Murasugi-Thistlethwaite Theorem, relating the Jones polynomial of an alternating link to certain refinements of the crossing number. Finally, we construct a spanning tree model for the Khovanov homology of nullhomologous links, mirroring work by Champanerkar-Kofman and Wehrli for links in $S^3$. For alternating links, we use our model to prove that the Khovanov homology in $\mathbb{Z}/2\mathbb{Z}$ coefficients is determined entirely by the Jones polynomial and signatures of the link.

We prove that the Kazhdan–Lusztig polynomial of a loopless sparse paving matroid is unimodal. When the rank is at least five and the corank is positive, we show that the coefficient sequence is strictly unimodal except for three explicitly determined parameter triples, each giving two equal maximal coefficients. We also determine the modes and classify the degenerate loopless sparse paving matroids of positive rank and corank.

We investigate a recently introduced generalisation of determined directions in affine planes. A direction $(d)$ in an affine plane is a point of the line at infinity in the projective completion, and hence corresponds to a parallel class. A (multi)set $S$ of points is equidistributed from direction $(d)$ if all lines from that parallel class intersect $S$ in the same number of points. In this paper, we give a construction of point (multi)sets that are inequidistributed from exactly 3 directions in any finite translation plane, and prove that all such (multi)sets arise from this construction, generalising a result of Kiss and Somlai in Desarguesian planes of prime order. We also use ideas from algebraic graph theory to prove results on directions associated with a pair of point sets $S$ and $T$. If for every direction $(d)$ either $S$ or $T$ is equidistributed from $(d)$, we prove that $|S \cap T| = |S||T|/q^2$, where $q$ is the order of the affine plane. In the appendix, we give a combinatorial alternative approach to proving this equality, which can be of independent interest. We also introduce the notion of directions cross-determined by a pair of sets $S$ and $T$, and prove that $|S| |T| \leq q^2$ if $(S,T)$ cross-determines at most half of the directions, where equality forces $S = T$.

We describe one explicit simple graph G on 32 vertices whose adaptive, normal, linear, entropic, and submodular widths are pairwise distinct. We compute all these widths exactly, except for the entropic width, where we only give a lower and upper bound. We use Ingleton's inequality and the Zhang-Yeung inequality for upper bounds, and give explicit constructions of modular, normal, linear, entropic, and non-entropic polymatroids for lower bounds.

We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\,\mathrm{per}(A_{S\cap T})\ge\mathrm{per}(A_S)\,\mathrm{per}(A_T)$ for all $S,T\subseteq[n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.

We enumerate all families of involutions avoiding a classical pattern of length four together with 4321. While the enumeration of permutations avoiding two classical patterns is by now well understood, the case of pattern-avoiding involutions presents additional structural constraints that require dedicated techniques. In particular our main enumerative tool is Biane's bijection between involutions and labelled Motzkin paths. The paper provides exact bivariate generating functions (taking into account length and number of descents), and structural descriptions for all except one family. In the remaining case we provide a functional equation for the generating function and express it as an explicit continued fraction. We make extensive use of experimental and software-supported methods.

We present a new lower bound on the number of lattice points of all totally unimodular polytopes, generalizing previous lower bounds on contingency tables, integer flows, and beyond. Our bound is based on the Gurvits capacity convex optimization problem, and thus our result implies an efficient deterministic algorithm for approximate counting of the lattice points up to an explicit exponential factor. We achieve our bounds by showing that the associated generating polynomials fit into a new general class of log-concave polynomials called VLC ("variable-wise log-concavity”). This also implies a conjecture of Ferroni and Higashitani on the evaluations of the Ehrhart polynomials of unimodular polytopes. The essential ingredient for these results is the resolution of Barvinok's log-concavity conjecture for contingency tables on lines, which was proven using ChatGPT 6 Astra. We conjecture a generalization of Barvinok's conjecture, which we believe will lead to stronger and more general bounds.

Routing unknown quantum information across distributed communication networks requires autonomous, measurement-free mechanisms to prevent wave-function collapse. Szegedy quantum walks provide a mechanism for spatial state transport. The conventional walks on unweighted graphs suffer from severe back-reflection, spatial dispersion, and channel crosstalk. In this paper, we introduce a deterministic topological quantum routing architecture based on coined Szegedy quantum walks on edge-duplicated signed graphs. By treating edge signs as localized phase shifts within a balanced coin reflection, we enforce an exact zero back-scattering condition across routing nodes. We demonstrate deterministic Perfect State Transfer (PST) with unit fidelity at exact arrival times across fundamental archetypes, including the signed dumbbell switch $D_{2m,0,2n}$ and scalable glued binary trees. Furthermore, we analyse routing performance under realistic open-system amplitude and phase damping noise channels.

Let $q$ be an odd prime power and $m\geq 1$. Let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^{2m}$. For every sphere $S_j=\{Q=j\}$ with $j\in \mathbb F_q^\times$, we prove the sharp extension estimate $R_{S_j}^*(2\to r)\lesssim_m1$ for $r\ge 2(m+1)/m$, uniformly in $q$, $Q$, and $j$. As an application, we show that if $E\subseteq\mathbb F_q^{2m}$ satisfies $|E|/q^{m+1/3}\to\infty$, then almost every pin $y\in E$ determines $(1-o(1))q$ values of $Q(x-y)$. The proof combines two arithmetic Hecke operator estimates with an induction in the dimension, a centered sphere–cone estimate, and an orthogonal decomposition.

For every fixed integer $d\geq2$, we prove that the finite graphs receiving a continuous homomorphism from the standard Schreier graph $F(2^{\mathbb Z^d})$ form a $\Sigma^0_1$-complete set. This extends a theorem of Gao, Jackson, Krohne and Seward when $d=2$. For the positive part of the reduction, we show that if a graph $H$ satisfies a certain property then there is a continuous graph homomorphism from $F(2^{\mathbb Z^d})$ to $H$, in particular, the complete graph $K_4$ satisfies this property. This extends a theorem of Gao and Jackson. We also prove that $\big (\mathcal H_d\big )$ is a strictly decreasing sequence.

When $G$ is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of $G$. In this paper, we prove a similar theorem for certain subclasses of rank 2 affine MV polytopes. For the Kac-Moody group $\widehat{SL_2}$, an affine MV polytopes splits into three subpolytopes: a lower, a middle and an upper polytope. The lower polytopes are natural generalizations of finite-type polytopes with highest vertex labelled by an arbitrary Weyl element. We extend the known results from this subclass of finite-type MV polytopes to the case of lower affine MV polytopes of rank 2. We prove that for an element $w$ of the affine Weyl group, the class of lower affine MV polytopes with highest vertex $w$ are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$. To do this, we describe the BZ data of a rank 2 affine MV polytope and show that certain generalized minor functions satisfy the conditions of a lower polytope. As any upper polytope is a reflection of some lower polytope, a analogous result will hold for the class of upper affine MV polytopes of rank 2.

Let $\mathcal{F}$ be a nonempty family of connected bipartite graphs, each with at least two edges. For a graph $G$, a proper vertex coloring of $G$ is $(2,\mathcal{F})$-avoiding if no member of $\mathcal{F}$ occurs bichromatically, and $\chi_{2,\mathcal{F}}(G)$ denotes the minimum number of colors in such a coloring. A B-coloring of $G$ is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in a B-coloring of $G$. For a fixed connected bipartite graph $F$ with at least one edge and bipartition classes $X_F$ and $Y_F$, define $k(F)=\min\{|I|:I\subseteq X_F\text{ or }I\subseteq Y_F,F-I\text{ is a forest}\}$. Let $m\ge2$ be the minimum number of edges in a member of $\mathcal{F}$. We prove that if $k(F)\le m-2$, then every $F$-free graph $G$ of sufficiently large maximum degree $\Delta$ satisfies $\chi_{2,\mathcal{F}}(G)=O((\frac{\Delta^m}{\log\Delta})^{\frac{1}{m-1}})$, which gives a positive answer to Chuet's Problem A and C in a sharp sense, thereby extending the results of Chuet [arXiv:2603.23379] from frugal colorings to $(2,\mathcal{F})$-avoiding colorings. For B-colorings, put $k=k(F)$, $h=|V(F)|$, and $s=\min\{|X_F|,|Y_F|\}$. We prove that every $F$-free graph $G$ of sufficiently large maximum degree $\Delta$ satisfies \[ q_B(G)\le \begin{cases} \Delta+\Delta^{1-\eta}+1, & \text{if }s\le2,\\ (4h-2)(\Delta-1)+1, & \text{if }s\ge3\text{ and }k\le1,\\ C\frac{\Delta^{2-\frac{1}{k}}}{\log\Delta}, & \text{if }k\ge2, \end{cases} \] where $\eta>0$ and $C>0$ depend only on $F$. For $k\le1$, the linear order is best possible, and for $k\ge2$, the bound is nearly sharp. To prove these results, we develop a common reduction of the coloring problems to $P$-perfect matching problems in auxiliary hypergraphs and apply the forbidden-submatching theorem of Delcourt and Postle.

We study the critical exponent of additive codes through an integer polymatroid associated with the code. We give a coding-theoretic proof of Whittle's Critical Theorem in this setting, a geometric description of the critical exponent in terms of $h$-projective systems, and general bounds, including an analogue of Kung's girth bound. We then study additive codes whose polymatroid is the hypergraphic polymatroid of a hypergraph $H$. For these codes the critical exponent turns to be determined by the weak chromatic number of $H$. If the code is faithful, then the minimum folded Hamming weight of the dual code is equal to the Berge girth of $H$. If $H$ is connected, the minimum distance is equal to the edge-connectivity of $H$. As a consequence, for $h\geq2$ we determine all such codes with connected $H$ that attain the Singleton bound, that is, all faithful hypergraphic additive quasi-MDS codes. We specialise the Griesmer and linear programming bounds to hypergraphic codes, we derive a lower bound on the minimum distance from the Laplacian eigenvalues of the weighted $2$-section of $H$, and we compare all these bounds computationally.

We study a support-collapse version of graph pebbling on paths. A configuration is stackable if a sequence of legal pebbling moves can produce a nonzero configuration supported on a single vertex. On the path P_n, we choose a configuration uniformly from all weak compositions of total n times mu_n, where mu_n is a positive integer. We prove a two-sided fixed-offset transition for the logarithmic density. The transition is centred at sqrt(log_2 n) - (1/2) log_2 log_2 n + log_2(3e). For every fixed epsilon greater than zero, the stackability probability tends to zero when log_2 mu_n is eventually at most the centre minus epsilon, and tends to one when it is eventually at least the centre plus epsilon. No assertion is made at zero offset. The proof uses an exact recursive stackability score on trees, a one-dimensional path-message recurrence, binary-partition asymptotics for rare dyadic deficit excursions, a constant-cost regeneration argument, and an exact deep-message necessity theorem. Conditioning independent geometric occupancies on their sum returns the uniform fixed-total model. The finite deterministic necessity theorem and its exact fixed-total corollary are formalised in Lean and registered with Palomar; the full probabilistic asymptotic theorem is not part of that registration.

A partition $\pi=\{V_{1},V_{2},...,V_{k}\}$ of the vertex set $V$ of a graph $G$ into $k$ color classes $V_{i},$ with $i\in\{1,...,k\}$ is called a quorum coloring if for every vertex $v\in V,$ at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v.$ The maximum cardinality of a quorum coloring of $G$ is called the quorum coloring number of $G$ and is denoted by $\psi_{q}(G).$ A sub-quorum coloring of $G$ is an onto partial function $f:V\rightarrow\left\{1,2,\ldots,\ell\right\}$ having the property that for every vertex $v\in V,$ if $f(v)$ is defined, then at least half of the vertices in $N[v]$ having an image by $f$, have the same color as $v.$ The sub-quorum coloring number $\psi_{sq}(G)$ equals the maximum value $\ell$ in a sub-quorum coloring of $G.$ In this paper, we determine the exact value of the sub-quorum coloring number for some infinite families of caterpillars including complete $n$-tuple caterpillars and complete caterpillars with minimum spine-vertex degree three.

The weak, stable, and ordinary Lusternik–Schnirelmann categories of a finite $T_0$-space $X$ satisfy $\operatorname{cat}_w(X)\leq \operatorname{cat}_s(X)\leq \operatorname{cat}(X)$. We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Villar-Liñán. If $P$ is weakly contractible but noncontractible and the deletion of one point makes $P$ contractible, then adjoining $m\geq 2$ incomparable maximal points produces a connected finite space with category triple $(1,2,m)$. Moreover, after any positive number of barycentric subdivisions its ordinary category is equal to $2$. Applying the construction to a nine-point space yields examples on $m+9$ points and, in particular, a twelve-point example for which both inequalities are strict. Using additivity under disjoint unions, we also realize every triple $(a,b,c)$ with $1\leq a<b\leq 2a$ and $c\geq b$, as well as every triple $(a,a,c)$ with $2\leq a\leq c$. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.

A metric space is plastic if every bijective nonexpansive self-map is an isometry. We prove that the hyperspace of nonempty compact subsets of every connected, locally finite, regular graph, equipped with the Hausdorff metric associated with the path metric, is plastic. The proof identifies the sets with smallest unit balls as the nonempty subsets of adjacent-twin classes and shows that every nonexpansive bijection induces an automorphism of the quotient graph preserving the sizes of these classes. Singletons are preserved when there are no adjacent twins, but need not be preserved in general. We also prove that $\mathcal{K}(K\times G)$ is plastic for every compact connected metric space $K$ and every connected, locally finite, regular graph $G$, with the supremum metric on the product, and give a more general criterion for such products. Further results include a plastic metric space whose hyperspace is not plastic, plasticity of every tree, and plasticity of the hyperspace of a connected, locally finite graph with only finitely many vertices of minimum degree.

A pseudohemisphere hypergraph is a hypergraph $\mathcal{H}$ with an ordered set of vertices $V$ for which there exists an $ABA$-free hypergraph $\mathcal{F}$ on $V$ and a subset $X$ of $V$ such that the hyperedge set of $\mathcal{H}$ is a subset of $\{F\Delta X: F\in \mathcal{F}\cup \overline{\mathcal{F}}\}$. We prove that the chromatic number of pseudohemisphere hypergraphs is at most four.

Let $W_s=K_1+C_{s-1}$ denote the wheel on $s$ vertices. We give structural proofs that $R(C_{14},W_{11})=27$ and $R(C_{15},W_{11})=29$. Together with the theorem of Chen et al. for $n\ge16$, these equalities give $R(C_n,W_{11})=2n-1$ for every $n\ge14$. The two boundary values were included in an earlier survey announcement. We also give structural proofs of $R(C_8,W_7)=15$, $R(C_9,W_7)=17$, and $R(C_8,W_9)=15$. The common starting point is a Hamilton-connected core lemma. For the eleven-vertex wheel, bounds on vertex connectivity and on the matching number of a bipartite graph associated with a local cycle yield a vertex cut of order nine. Paths with prescribed endpoints then rule out every possible pair of orders of the two remaining vertex sets. For the smaller wheels, we use the structure of critical cycle colorings and local cycle-shortening arguments. We also give complete structural classifications of the $(C_8,C_6)$- and $(C_9,C_6)$-critical colorings, recovering the previously reported counts 24 and 26. All proofs are combinatorial and use no exhaustive graph enumeration.

We study affine deformations $\mathcal{A}(G_{\mathcal{S}})$ of graphic arrangements and their cones. Here $G=([n],E(G))$ is a simple graph, $\mathcal{S}=(S_{ij})$ is a family of finite gain sets, and $\mathcal{A}(G_{\mathcal{S}})$ consists of the hyperplanes $x_i-x_j=a$ with $a\in S_{ij}$. We call $G_{\mathcal{S}}$ blockwise admissible if every block has a vertex ordering $v_1,\ldots,v_m$ satisfying $S_{v_kv_j}-S_{v_kv_i}\subseteq S_{v_iv_j}$ for every $k$ and all distinct $i,j>k$. Every such ordering is a perfect elimination ordering. We prove, for arbitrary $G$, that the following are equivalent: (i) $G_{\mathcal{S}}$ is blockwise admissible; (ii) the cone $c\mathcal{A}(G_{\mathcal{S}})$ is supersolvable; and (iii) $\mathcal{A}(G_{\mathcal{S}})$ admits a nice partition. We give a direct arrangement-theoretic proof of these equivalences. For a block, the equivalence between admissibility and supersolvability is already contained in Zaslavsky's characterization of supersolvable graphic-lift lattices. We further show that every nice partition on a block is induced by an admissible ordering and that its induced edge classes are stars with distinct centers. Under these equivalent conditions, we construct a maximal modular chain through the hyperplane at infinity. We also prove that if $c\mathcal{A}(G_{\mathcal{S}})$ is free, then $G$ is chordal.

We investigate the structural, combinatorial, ideal-theoretic and Krohn-Rhodes complexity of the Margolis-Rhodes monoid, MR(G), of a finite simple graph G, viewed topologically as a 1-dimensional simplicial complex. Alongside the full monoid, we examine some subsemigroups including St(G), defined by the condition that the full inverse image is an edge or the empty set and Inj(G), the monoid of all partial 1-1 continuous functions. We provide explicit combinatorial enumerations and struture for paths and cycles. We compute Green's relations showing in particular that the partial order of regular J-classes is isomorphic to the poset of induced subgraphs of G. Finally, we apply these structural invariants to Krohn-Rhodes complexity theory. It is known that the Margolis-Rhodes monoid has complexity at most 2 and 1 for St(G) and Inj(G). We show that for cycles the complexity of its Margolis-Rhodes monoid is 2 if and only if the cycle is of length at least 4. For paths, we prove that the complexity of its Margolis-Rhodes monoid is 2 if the path length is at least 13.

Let $G$ be the graph on the positive integers in which distinct vertices are adjacent if they differ by one or their quotient in one order is prime. We prove that $6\leq\operatorname{diam}(G)\leq7$ and $4\leq\operatorname{rad}(G)\leq5$. Under Dickson's conjecture for two linear forms, we prove that $\operatorname{diam}(G)=6$ and $\operatorname{rad}(G)=4$.

We present multitable and its Rust reference implementation: a stable hash table both materially faster at equal physical memory and more flexible than the SwissTable in its Rust's hashbrown implementation. As an arithmetic mean over 84 configurations it delivers $\mathbf{2.1\times}$ hashbrown's throughput when both hash the same raw bytes and $\mathbf{1.9\times}$ when hashbrown is keyed on native integers, its best case; on negative lookups alone, $3.2\times$ and $2.9\times$. Multitable reaches any physical load factor up to and including one ($0.9999$ demonstrated), exactly for the requested capacity, compared to hashbrown which doubles at $0.777$ for 4-byte keys and values. At $75\%$ saturation of hashbrown (assumed average case of its rigid ladder) and multitable sized to $0.97$ physical load factor, hashbrown takes $66\%$ more space. The lookup probe count has no cliff as the load factor approaches one. Bucket size, physical load factor, and failure budget are parameters, and the multitable can be grown without rehashing. We implement two variants of multitable: plain and filtered. At equal physical memory on an Apple M2 Pro the filtered multitable leads hashbrown in all $84$ insert, hit, and miss configurations. Multitable is more memory-efficient, at equal mixed-lookup throughput on the map of $4$-byte keys and values the filtered multitable needs up to $12\%$ fewer bytes than hashbrown, and the plain multitable is $18\%$ smaller, holding $\mathbf{22\%}$ more keys in the same memory.

We develop a multi-view linear-programming framework for locally recoverable codes with arbitrary fixed availability $a$. For any retained order $1 \le s \le a$, the selected helper sets, the recovered coordinate, and their complement form an $(s+2)$-part partition. Recording the Hamming distance on all blocks preserves both compatibility among the selected repair alternatives and their coupling with the remaining coordinates. The resulting joint distribution satisfies centered counting identities, product-Krawtchouk positivity, and collision inequalities from the local-distance condition; for fixed $s$, these constraints give a polynomial-size relaxation for arbitrary, possibly nonlinear, codes over any finite field. Retaining one view recovers the three-block model of our companion paper. We develop the first genuinely multi-view case, $s=2$, in detail and specialize the exact computations to availability $a=2$. The resulting four-block LP projects to both the one-view three-block model and a globally conditioned two-view relaxation, making explicit the information lost by each coarsening. Exact rational primal-dual certificates together with checked constructions prove $M_{\max}(2,8,4,2,2,2)=8$, $M_{\max}(3,7,3,2,2,2)=27$, and $M_{\max}(4,7,3,2,2,2)=64$; in all three cases the four-block bound is strictly stronger than both coarsenings.