Papers

Let $q$ be a power of an odd prime $p$. We prove the MDS conjecture over $\mathbb F_q$ in every dimension $k$ satisfying \[ 2\leqslant k\leqslant B(p,q) \quad\text{or}\quad q+2-B(p,q)\leqslant k\leqslant q, \qquad B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor. \] For fixed $p$, the first interval is linear in $q$; together with duality, the theorem covers an asymptotic proportion $1-1/(2p-3)$ of all dimensions. The proof rests on a vanishing theorem for determinant relations over an arbitrary field of characteristic $p>0$. It yields full row rank for Chowdhury's matrices over a larger range of arc sizes. In particular, at $|G|=2k-3+n$ it proves Chowdhury's full-row-rank conjecture without the $q$-dependent restriction. Specialization to $\mathbb F_q$, together with the Ball–Lavrauw construction, gives the stated MDS range. We also prove that, for every odd prime power $q$, every normal rational curve in $\mathrm{PG}(N,q)$ is complete for $2\leqslant N\leqslant q-2$, and every projective Reed–Solomon code of length $q+1$ and dimension $2\leqslant k\leqslant q-2$ has covering radius $q-k$.

We give a proof that the full group $C^*$-algebra of a countable discrete amenable group is strong NF. The representation-theoretic construction uses the exact sequence whose kernel is the hyper-FC radical and whose quotient is the maximal ICC quotient. Extreme invariant traces on the radical induce factorial representations of the group. A Choquet decomposition of the regular radical trace selects a separating family whose induced traces are supported on the ordinary FC-centre. Exact twisted crossed-product quotients and a sequential UCT argument make their images quasidiagonal. Faithful irreducible replacements then supply the projections required for inner quasidiagonality. We also give a direct trace-decomposition variant and identify the steps shared by the two proofs.

We construct a separable unital residually finite-dimensional $C^*$-algebra $A$ whose maximal tensor square contains a proper isometry. In particular, $A$ is quasidiagonal whereas $A\tmax A$ is not. The algebra is generated by finite regular representations obtained from congruence quotients of an explicit countable affine group. A property~$(T)$ subgroup determines a diagonal Kazhdan projection in the maximal tensor square. Conjugation by a dilation makes this projection equivalent to a proper subprojection. Strictness is detected by a quasi-regular representation whose two factor restrictions descend to the specified algebra $A$. We give the norm estimate needed for this descent and construct the proper isometry explicitly.

We characterize the fresh-gradient oracle complexity of smooth nonconvex-strongly-concave minimax optimization, with matching upper and lower bounds up to logarithmic factors. Let $\Phi(x)=\max_y f(x,y)$, where $f$ is jointly $L$-smooth and $\mu$-strongly concave in $y$ on unconstrained Euclidean domains, and set $\kappa=L/\mu$. Each query returns a fresh unbiased joint gradient with conditional variance at most $\sigma^2$. Given initial primal gap at most $\Delta$ and dual residual $\|\nabla_y f(x_0,y_0)\|\le G$, the fixed-budget complexity of finding $\|\nabla\Phi(\widehat x)\|\le\epsilon$ with probability at least $2/3$ is $\widetilde{\Theta}(\sqrt{\kappa}L\Delta/\epsilon^2+\kappa L\Delta\sigma^2/\epsilon^4+\kappa^2\sigma^2/\epsilon^2+\sqrt{\kappa}\log_+(G/(\epsilon\sqrt{\kappa})))$, where $\log_+u=\log\max\{1,u\}$. This characterization holds when $\kappa$ and $L\Delta/\epsilon^2$ exceed universal constants, uniformly over finite dimensions, and the suppressed logarithms are independent of $G$. It establishes the necessity of linear condition-number dependence in the global stochastic cost and identifies a separate statistical refinement cost. A proximal method separates coarse primal progress from one final refinement, while the lower bounds apply to arbitrary adaptive randomized algorithms. Dual initialization contributes only an additive logarithmic cost, yet removing its control eliminates every finite dimension-free complexity bound, even with exact gradients.

We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh–Sibony's list. For a saturated holomorphic curve foliation on a compact Kähler manifold with finitely many singularities, every positive directed $dd^c$-closed current has finite Poincaré mass. Thus linearizability is not needed for the geometric ergodic theorem in this setting. We construct a projective surface with dense hyperbolic leaves and a diffuse extremal current whose Nevanlinna averages converge at a uniform reciprocal logarithmic rate. This is an existence result for a chosen foliation, not a rate theorem for an arbitrary prescribed foliation. Under the hypotheses of the first result, every hyperbolic leaf has a logarithmic lower bound for its Nevanlinna characteristic, even if no diffuse directed current exists. Positive-dimensional singular sets are not treated.

We prove a capacity criterion for fixed-degree spatially coupled MacKay–Neal codes on additive symmetric Markov noise. The noise bit flips with probability $p$, so the channel has one parameter and capacity $1-h_2(p)$. For every integer $\ell\geq4$ with $3/\ell<1-h_2(p)$, matched finite-window BCJR and sum-product decoding admit code sequences, possibly depending on $p$, with actual rate tending to $3/\ell$ and vanishing average information-bit error. The proof parallels the GEC-to-BEC argument: a conditional-entropy correction transfers BMS fixed-point positivity to the channel with memory. The correction also covers a capacity deficit of the effective BMS channel. A two-state posterior contraction supplies the additional derivative bound needed for threshold saturation. The result uses the cited BMS positivity theorem, with an analytic proof for $\ell\geq33$ and exact interval certificates for $4\leq\ell\leq32$.

We study the cohomology of iterated Kummer extensions on the moduli space of five-pointed rational curves, continuing [arXiv:2609.19341, arXiv:2609.21356]. A two-dimensional shrinking construction compares the Betti presentations obtained from its five forgetful projections. The corresponding Hodge defects give pentagram relations, quadratic relations between coefficients of the Drinfeld associator, which are equivalent to the pentagon equation under group-likeness, normalization, and reversal. Under reversal and vanishing of pure-power coefficients, the relations imply shuffle and regularized double shuffle, whose Hodge realization is recovered by a convolution specialization.

Generating functions in curve counting theory mix two different kinds of information: multiple covers mix curve degrees, while the topological string coupling mixes BPS genera. We show that a single central difference operator separates these two effects in a precise and invertible way for the all genus Gopakumar-Vafa expansion along an effective ray of a Calabi-Yau threefold. Writing \[ y=\left(2\sin\frac{\lambda}{2}\right)^2, \] we combine the Bryan-Pandharipande cover polynomials with the corresponding Fejér factors and obtain a Chebyshev-Fejér ghost transform for the normalized difference forcing. The transform admits an explicit operator-valued Möbius inverse. Its $y$-adic filtration is exactly the BPS genus filtration: the forcing modulo $y^{G+1}$ is equivalent to the complete collection of BPS invariants of genera at most $G$. Thus the genus zero reconstruction from the classical specialization is the first associated graded piece of an all genus hierarchy. We also characterize the integral image of the transform. Integral BPS forcing is equivalent to a two stage condition: Witt-Dwork type Frobenius congruences for the underlying ghost coordinates, together with an additional Fejér divisibility condition imposed by the central difference operator. In prime power degree this gives a strengthened Dwork congruence containing the full Fejér factor. For local curves, the formalism has concrete geometric realizations. A super-rigid elliptic curve gives an exact Dedekind $\eta$ expression for the forcing. In the balanced higher genus local curve theory, the upper degree two BPS invariants are realized as top Chern numbers on natural stacks of square roots over symmetric products. These results connect central difference equations with BPS reconstruction, arithmetic ghost structures, and the geometry of local curves.

Naryshkin recently proved that a free action of a countably infinite discrete amenable group $G$ on a compact metrizable space $X$ with the uniform Rokhlin property and comparison (URPC) admits an equivariant embedding into $([0,1]^m)^G$ whenever $\mathrm{mdim}(X,G)<m/2$. We strengthen this result from existence to genericity by a substantially different method: the set of observables $f\in C(X,[0,1]^m)$ whose orbit maps are equivariant embeddings into $([0,1]^m)^G$ is a dense $G_\delta$ subset of $C(X,[0,1]^m)$. The proof combines an adaptation of the Kakutani skyscraper construction used by Gutman–Tsukamoto with a new polynomial perturbation method, departing from the predominantly linear perturbation approaches used in earlier equivariant embedding literature.

We study oscillation of the second-order delay equation \[ x''(t)=x(\tau(t)), \] with a nondecreasing delayed argument $\tau(t)\le t$ satisfying $\tau(t)\overset{t\to\infty}{\to}\infty$. Set \begin{equation*} a=\liminf_{t\to\infty}(t-\tau(t)),\qquad b=\limsup_{t\to\infty}(t-\tau(t)). \end{equation*} We describe a continuous strictly decreasing function \begin{equation} \beta:[0,2/e]\longrightarrow[2/e,\sqrt2],\qquad \beta(0)=\sqrt2,\qquad\beta(2/e)=2/e, \notag \end{equation} such that, if the equation has an eventually positive decreasing solution, then \begin{equation} a\leq2/e,\qquad b\le\beta(a). \notag \end{equation} The bound $b\le\beta(a)$ is sharp and interpolates between the classical $\limsup$ threshold and the $\liminf$ autonomous threshold. A detailed spectral analysis gives a simple interpretation of the result and links it directly to the oscillation theory of first-order delay equations.

We extend the spherical-integral approach to $\mathcal{R}$-transforms of non-Hermitian random matrices to random matrices with a fixed $K \times K$ block structure. We introduce a rank-$K$ spherical integral defining a scalar $H$-transform on $2K \times 2K$ overlap matrices and an associated matrix-valued $\mathcal{R}$-transform encoding the joint block structure. Using the replica method, we derive a conjectural subordination relation for sums of independent block matrices. This framework provides a method for determining the spectral boundaries of large non-Hermitian random matrices with a fixed block structure.

Let $(\mathcal{X},\mathcal{Y})$ be an admissible balanced pair in a locally presentable abelian category $\mathcal{A}$, and let $\mathcal{E}$ be its induced exact structure. We study the class $G(\mathcal{X})$ of cycles of two-sided complexes with terms in $\mathcal{X}$ that are both right and left $\mathcal{X}$-acyclic. If $\mathcal{X}$ is closed under direct summands and finite direct sums and is $\kappa$-Kaplansky for every sufficiently large regular cardinal $\kappa$, we prove that $(G(\mathcal{X}),G(\mathcal{X})^{\perp *})$ is a hereditary cotorsion pair relative to $\mathcal E$. For relative purity determined by finitely presented modules, we give two criteria for functorial completeness and show that relative periodic modules generate a functorially complete hereditary cotorsion pair. We also compute the relative Gorenstein objects, their right orthogonal, and explicit approximations for a family of exact structures over Dedekind domains.

This paper studies weak mixing for two classes of dynamical systems with the \(\mathscr{M}_{\alpha}\)-shadowing property using the generic factor characterization of weak scattering. First, for every \(\alpha\in[0,1)\), every \(M\)-system with the \(\mathscr{M}_{\alpha}\)-shadowing property is weakly mixing, thereby answering [P. Oprocha et. al., On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl., Question~1] negatively. Second, for every \(\alpha\in(0,1)\), the \(\mathscr{M}_{\alpha}\)-shadowing property implies weak mixing for \(E\)-systems and, in particular, for dynamical systems whose measure center is the whole space.Finally, an \(E\)-system with the \(\mathscr{M}_{0}\)-shadowing property which is not weakly mixing is constructed, showing that the parameter range \(\alpha\in(0,1)\) in the \(E\)-system result is optimal.

We study rainbow spanning configurations in bijumbled graphs whose edges are coloured independently and uniformly from a prescribed palette. For $n$-vertex $(p,\beta)$-bijumbled graphs with minimum degree at least a fixed positive multiple of $pn$, we obtain rainbow perfect matchings and Hamilton cycles with a sufficient palette surplus of order $(\log n)/p$, assuming $pn=\omega(\log n)$ and $\beta\le cpn$ for a sufficiently small constant $c$. For each prescribed spanning tree of fixed maximum degree $\Delta\ge2$, a surplus of order $L_{n,\Delta}(\log n)/p$ suffices under $\beta\le cpn/L_{n,\Delta}$, where $L_{n,\Delta}=\Delta^{5\sqrt{\log n}}$. The palette surplus is sublinear under these hypotheses. These results use a McDiarmid-type coupling and retain the discrepancy scales of the relevant deterministic embedding theorems. We also prove exact-palette results, using precisely as many colours as the number of edges of the target configuration. After independent edge percolation at rate $\rho$, it is shown that a rainbow perfect matching or Hamilton cycle exists asymptotically almost surely when $\rho pn \ge C(\log n)^2$ and $\beta\le\gamma pn$ for appropriate constants $C$ and $\gamma$. We obtain corresponding results for each prescribed bounded-degree spanning tree and for clique factors under appropriate stronger hypotheses. The exact-palette proofs construct spread measures from uncoloured containment estimates and apply the rainbow threshold theorem of Han and Yuan.

In the literature, several graph matrices based on vertex degrees have been introduced, and the study of their spectral properties has attracted considerable attention in recent years. Motivated by these developments, in this paper, we investigate the spectral radius of the weighted adjacency matrix $A_f(G)$, where the function $f$ satisfies property $P^{*}$. More precisely, we characterize the graphs that attain the maximum spectral radius of $A_f(G)$ among graphs with a vertex cut set of prescribed order, graphs with exactly $s$ cut edges, and graphs with given vertex connectivity and independence number. Our results further develop the unified framework for studying extremal spectral properties of degree-based weighted adjacency matrices and extend this framework to several classes of graphs with prescribed structural parameters.

We analyze a simple stochastic inertial Krasnosel'skii–Mann (iKM) method for finding a fixed point of a nonexpansive operator in a real Hilbert space. Our method is obtained simply by adding two inertial extrapolations to stochastic KM [Bravo and Cominetti, 2024], and it retains one call to a possibly biased stochastic oracle per update and achieves sharp rates in both the stochastic and deterministic regimes. Specifically, with our proposed parameter schedule, we prove the following last-iterate fixed-point residual bound: \[ {O}\!\left(\frac{1}{K} +\frac{\sigma\log K}{\sqrt K} +\frac{B_K\log K}{K}\right), \] where $K$ is the horizon, $\sigma$ is the noise level and $B_K$ is the accumulated root-mean-square bias. When $B_K=O(\sqrt K)$, this yields $\widetilde O(\epsilon^{-2})$ sample complexity that matches, up to a logarithmic factor, the stochastic-oracle lower bound given under the unbiased subclass of our model [Foster et al., 2019, Theorem 2]. It also improves the best-known $O(\epsilon^{-4})$ random-iterate guarantee for stochastic KM [Bravo and Cominetti, 2024, Corollary 5.4]. To our knowledge, this is the first single-loop method for general nonexpansive fixed-point problems to attain this near-optimal sample complexity without variance reduction or batching. When the oracle is exact, the same method attains the worst-case-optimal $O(K^{-1})$ last-iterate residual rate [Park and Ryu, 2022, Theorem 4.6], improving the $O(K^{-1/2})$ rate of classical KM [Cominetti et al., 2014; Bravo and Cominetti, 2018].

We show, analytically, that a static sine-Gordon soliton cannot exist in 1 + 1 non-dynamical de Sitter spacetime if $\alpha := (m/H)^2 < 2$, where $m$ is the mass parameter of the sine-Gordon theory and $H$ is the Hubble constant. Conversely, we also show that static sine-Gordon solitons exist in 1 + 1 non-dynamical de Sitter spacetime if $\alpha > 2$. The above threshold is explained—qualitatively and to within an O(1) factor—using a heuristic argument involving the interplay of tensile force in the Lorentzian sine-Gordon soliton and the tidal force in de Sitter spacetime. A similar heuristic argument, which remains to be confirmed analytically, also suggests the existence of a threshold, $ (m_V/H)^2 \sim O(1)$, below which the tidal forces are too strong to permit the existence of a static `t Hooft–Polyakov monopole in non-dynamical 3 + 1 de Sitter spacetime; $m_V$ is the mass of the vector boson. Linde has suggested that new inflation could have triggered secondary inflation at the core of a GUT (Grand Unified Theory) monopole even if the Hubble constant at or after the GUT phase transition was significantly smaller than the mass of the $X$ boson. We present a heuristic argument, which suggests that the $SO(3)$ `t Hooft–Polyakov monopole does not allow secondary inflation at its core when the inflationary background is weak. Based on the above, as yet analytically unconfirmed, heuristic argument for the $SO(3)$ `t Hooft–Polyakov monopole, we conjecture that secondary inflation at the core of a GUT monopole is infeasible.

For type $A$ braid varieties, including Richardson varieties and positroid varieties, we prove that the standard torus has a dense orbit exactly when the associated link is smoothly slice. More generally, we relate the smooth slice genus to the codimension of a generic standard-torus orbit. For knots, this codimension equals twice the slice genus. We also characterize algebraic tori among open affine Richardson varieties and positroid patches by 2-crown avoidance in their Bruhat intervals. Finally, positroid links have equal Seifert and slice genera, so a positroid link is smoothly slice exactly when it is an unlink.

A weak form of the Hellinger conjecture of Anantharam, Bogdanov, Chakrabarti, Jayram, and Nair for the binary symmetric channel is proved: dictator functions maximize Hellinger $\Phi$-entropy among all Boolean functions of the input and all one-bit statistics of the output of a noisy channel. The technical heart of the matter is an explicit inequality in three real parameters, which is proved using explicit polynomial approximations and computer-assisted positivity checks. The results are also formally verified in Lean 4.

For an infinite connected locally finite undirected graph with positive symmetric edge weights and normalized Laplacian $\Delta$, we prove weak type $(1,1)$ estimates for two Riesz transforms. The edge transform has universal constant $2$, with no volume assumption. The vertex carré-du-champ transform has constant $H_1+1$ whenever $H_1=\sup_x m(B(x,1))/m(x)<\infty$. These endpoint constants apply to real-valued inputs and yield strong $\ell^p$ bounds for $1<p\le2$. The proof uses a discrete obstacle decomposition obtained by a contraction mapping, followed by elementary support estimates for the two gradients.

We establish a structural rigidity principle for the nontrivial zeros of the Riemann zeta function by analyzing the asymptotic interaction between discrete Dirichlet oscillations and their continuous analytic envelope. The starting point is a renormalized discrepancy operator, the C-transformation, which isolates the finite structural difference between the Dirichlet series and its integral approximation. Applied to the kernel x^(-s), this operator yields a decomposition of the zeta function into three parts: an oscillatory Dirichlet component X_N(s), an explicit analytic growth envelope Y_N(s), and a polynomially decaying remainder R_N(s). We show that the vanishing condition zeta(s)=0 forces an asymptotic compatibility between the two dominant components. This relation is formalized in a Bridge Lemma, which proves that zeros require asymptotic equality of the squared magnitudes |X_N(s)|^2 and |Y_N(s)|^2. A detailed asymptotic analysis then reveals that the ratio of these magnitudes exhibits three distinct regimes depending on the real part sigma=Re(s): divergence for sigma>1/2, collapse for sigma<1/2, and exact balance for sigma=1/2. This incompatibility of growth regimes produces a rigidity mechanism: the asymptotic balance required by the zero condition can occur only on the critical line. Consequently, every nontrivial zero must satisfy Re(s)=1/2. The result provides a deterministic explanation for the location of the zeros, showing that the critical line emerges from an intrinsic structural constraint of the Dirichlet representation of the zeta function.

Kahn proved that, among bipartite $d$-regular graphs on $n$ vertices, the number of independent sets is maximized by a disjoint union of copies of $K_{d,d}$. Zhao later extended this result to all $d$-regular graphs. We prove a robust version of this theorem in which independent sets are replaced by sets spanning few internal edges. If $G$ is $d$-regular on $n$ vertices, then the number of subsets spanning at most $\gamma dn$ edges is at most $$ 2^{n/2}\exp\left\{ O\bigl(\gamma\log(e/\gamma)n\bigr) + O\bigl(n/d\bigr) \right\}. $$ Both correction terms are sharp up to absolute constants: the $\gamma\log(1/\gamma)n$ term is necessary when $d$ is sufficiently large in terms of $\gamma$, while the $n/d$ term is already necessary for independent sets. Our result answers a question of Seth.

We quantify the entropy-decrement proof of the logarithmically averaged two-point Chowla theorem, uniformly in the affine forms and the averaging window. For L_i(n)=a_i n+b_i with positive integral slopes and nonzero determinant Delta=a_1 b_2-a_2 b_1, put h=max(a_1,a_2,|b_1|,|b_2|,|Delta|,2). For either the Liouville or Mobius function in each position and 0<epsilon<=1/10, the correlation sum over x/omega<n<=x, weighted by 1/n, has absolute value at most epsilon log omega whenever x>=omega>=exp_4(C h epsilon^(-2)), where exp_4 denotes four iterated exponentials. A translation lemma separates the common translation from the tower: for lambda(n+b)lambda(n+b+k), with k nonzero, cancellation is uniform when |k|=o(log_4 omega) and |b|=omega^(o(1)), where log_4 is the fourth iterated logarithm. The same thresholds give logarithmic equidistribution of the four Liouville sign pairs. The constant C is effective but depends on uncomputed constants in two analytic inputs. Separately, writing ell=log log x, we prove fully explicit bounds ell/3-(4/3)log ell-7 <= N_lambda(x) <= ell-log ell+2.53/ell for ell>=64, where N_lambda is the non-pretentiousness level in Tao's hypothesis. Under GRH the lower bound improves to ell-log ell-log 2-2/ell; hence N_lambda(x)=ell-log ell+O(1). The arguments use an explicit zero-free region, Harnack's inequality, a finite resonator, and a comparison of smoothed Euler products. These results concern logarithmic averages, not natural-density two-point Chowla correlations.

We prove that the function $$ \log \frac{\Gamma(x+a+b)\Gamma(x)}{\Gamma(x+a)\Gamma(x+b)}-b\log\left(1+\frac{a}{x}\right) $$ is a generalized Stieltjes function of order $3$ for $a>0$ and $0<b<1$. A similar result is obtained when $a>0$ and $b>1$.

We characterize boundedness of the Bergman projection on operator-weighted $L^p$ spaces over a separable Hilbert space, for $1<p<\infty$. Under the directional integrability conditions, the criterion is uniform boundedness of averages over Carleson squares in the unit disc. We obtain norm estimates independent of dimension. The proof uses a separated kernel expansion and a layer decomposition of the tent tree. No reverse Hölder assumption is needed. We also give a finer matrix estimate with a dimension-dependent constant. Boundedness is unchanged by compactly supported changes of the weight that preserve the integrability conditions.

We establish a cutoff criterion for finite Markov chains with non-negative Bakry–Émery curvature without assuming that the support of the transition matrix is symmetric. This extends a criterion of Salez based on a refined product condition. For each allowed transition, we consider a shortest directed path from its endpoint back to its starting point. Our cutoff criterion depends on the maximum length of these paths. We also obtain a discrete-time counterpart for lazy chains. For random walks on finite Abelian groups, we prove non-negative Bakry–Émery curvature for arbitrary generating sets. As an application, we establish almost-sure cutoff for random walks on products of cyclic groups of order three.

In this paper, we establish a Bourgain–Brezis–Mironescu formula for BV functions on arbitrary open sets for a general class of radial mollifiers. Building on this formula, we identify the \(\Gamma\)-limit of the improved nonlocal energies with respect to \(L^1\)-convergence as a constant multiple of the total variation. As a consequence, we obtain a characterization of BV functions in terms of the finiteness of the corresponding nonlocal energies. We also establish local \(L^1\)-compactness on arbitrary open sets and global \(L^1\)-compactness on bounded open sets under a natural uniform integrability assumption. In particular, except for the global compactness result, all these results hold on arbitrary open sets, with no assumptions on boundedness, connectedness, or boundary regularity. Counterexamples show that for global compactness, both the boundedness of the domain and the uniform integrability assumption are essential. The proofs rely primarily on one-dimensional restrictions of BV functions and a concentration argument for radial mollifiers.

Recently, Heath, McCourt, Parker, Schwieder, and Zerbib initiated the systematic study of the $r$-Kneser Ramsey number $R_r^{KG}(s,t)$ and investigated related Ramsey-type problems. A central motivation for their work comes from a question of Holmsen, Hrusak, and Roldán-Pensado, who asked whether, for $n=2k-1$ and sufficiently large $k$, every red/blue edge-coloring of the complete graph on the vertex set $V = \binom{[n]}{k}$ necessarily contains a monochromatic triangle $ABC$ with $A,B,C \in V$ and $A \cap B \cap C = \emptyset$. Heath, McCourt, Parker, Schwieder, and Zerbib established that this conclusion holds when $n \ge \frac{7k}{3}$ and $k\ge 12$. We make substantial progress toward the problem of Holmsen, Hrusak, and Roldán-Pensado by proving that the conclusion already holds for every $k\ge 2$ whenever $n\ge 2k+1$. In addition, we obtain improved lower bounds for $R_r^{KG}(s,t)$ when $s$ and $t$ are fixed and $r$ is sufficiently large.

We study the Ooms multiplicities of Frobenius maximal parabolics $W(a,b)$ of type~$A$. For $a=mb+r$ with $1\le r<b$, the Euclidean algorithm relates their potential histograms to those of the smaller algebra $W(r,b-r)$. We prove strict unimodality for every positive coprime pair: the multiplicities increase strictly to the equal central values at eigenvalues $0$ and $1$, then decrease strictly. This strengthens the recent maximal-parabolic unimodality theorem of Giaquinto, Irving, Lauve, and Mastnak. We also prove log-concavity for the unbounded families $W(mb\pm t,b)$, where $t\in\{1,2,3\}$, $b>t$, $\gcd(b,t)=1$, and $m\ge1$. These results cover every Frobenius case with smaller block at most $8$, and also smaller block $10$. For each fixed residue class, we then prove that log-concavity of the whole family is determined by finitely many initial spectra. The finite cutoff is obtained symbolically. Exact integer verification of the resulting finite lists proves strict internal log-concavity whenever the smaller block is at most $128$, with no bound on the larger block. The unrestricted log-concavity conjecture is not proved here.

We prove a sparse version of the four-colour theorem of Brandt and Thomassé, answering a question of Allen, Böttcher, Kohayakawa and Roberts. For every fixed $0<\gamma\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $\delta(H)\ge(1/3+\gamma)pn$ can be made four-partite by deleting at most $\min\{C_\gamma n/p,(1/8+\gamma)pn^2\}$ edges. In fact, deleting at most $C_\gamma n/p$ edges yields a graph that admits a homomorphism to an Andrásfai or Vega graph with certificate complexity at most $1/(3\gamma)$. Together with matching lower bounds from random blow-ups, this structural result determines, uniformly in $p$, the minimum-degree thresholds for $q$-partiteness with $O(n/p)$ edge deletions: $2/5$ for $q=2$, $10/29$ for $q=3$, and $1/3$ for every fixed $q\ge4$. For every fixed $q\ge2$ and $\log n/n\ll p\ll n^{-1/2}$, asymptotically almost surely $G(n,p)$ contains a spanning triangle-free subgraph with minimum degree $(1-o(1))pn$ that requires $(1/(2q)+o(1))pn^2$ edge deletions to become $q$-partite, showing that the coefficient $1/(2q)$ cannot be improved even under this stronger degree condition.