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We study the completion time of a task subject to independent reset (catastrophe) at each step. The completion-time PGF depends on the base-process PGF through an affine relation, and we exploit this structure systematically. Our main result shows that, among age-based catastrophe mechanisms, geometric-tail catastrophe is exactly the class that yields uniform affine PGF structure; in continuous time, the characterization sharpens to Poisson resetting. We establish a sharp two-sided Kolmogorov bound of order $p+|\alpha|$ for the exponential approximation $d_K(T/E[T], \mathrm{Exp}(1))$, thereby closing a logarithmic gap. Applications to the coupon collector with reset coupons reveal a discontinuous Gumbel-to-Exponential transition under resetting, while a multi-phase model exhibits a Gaussian-to-exponential transition with exponential convergence rate.

We study how far a fixed Ramsey upper bound can be improved by descending through blue neighborhoods in one vertex set while keeping a second set fixed. A weighted inequality in the two set sizes determines when the descent can stop. For the source bound specified here, the infimum diagonal exponent over all finite derivations lies in $[1.305,\,1.307]$. A finite derivation gives $R(k,k)\le3.69507^k$ for all sufficiently large $k$; a concave polygon proves the lower bound for every finite depth. We also characterize the infimum as a greatest fixed point and show that every larger exponent has a finite derivation valid uniformly for nearby clique-size ratios.

Isotonic regression and mean estimation over a simplex are among the most classical denoising problems under geometric constraints. Chatterjee and Lafferty generalized both to the recovery of a flow on a rooted tree from Gaussian noise, with isotonic regression on a path and the simplex on a star. Their results reach only special trees; on a general tree the minimax risk remained unknown for a decade. We settle the question in full. On every tree with $n$ vertices, at every budget $V$ and noise level $\sigma$, the minimax risk is of order $\min\{V^2H_T,\ \sigma^2k_0\}$, with $H_T$ the diameter and $k_0$ the crossing index of a truncated ancestor profile. A deterministic estimator, an exponentially weighted aggregate over integer states, attains it in $O(n^2\log(n+1))$ arithmetic operations. Least squares at a known budget, the natural convex program for the model, is suboptimal in the worst case by a factor of order $(\log n)^{2/5}$. One functional, the truncated ancestor profile, carries the rate, the estimator, and the least squares comparison. It serves as covering radius, as packing entropy in the positive cone, and as the range of the one-dimensional state behind the exact computation. Two classical theories become one on trees, sharp and computable throughout.

Let $\nu(G)$ be the maximum number of edge-disjoint triangles in a graph $G$ and $\tau(G)$ the minimum number of edges meeting every triangle. Tuza conjectured that $\tau(G)\le 2\nu(G)$. We prove that $\tau(G)\le (165/59)\nu(G)$. The constant $165/59\approx 2.797$ improves the bound $66/23\approx 2.870$ that Haxell proved in 1999. The key observation is that, for a suitable red-blue coloring, the families left over in Haxell’s construction contain every triangle with exactly one red edge. Such a family $\mathcal{F}$ admits an exchange that forces certain red edges to lie in a single triangle once the blue edges of a maximum packing are deleted, which gives $\tau(\mathcal{F})\le (8/3)\nu(\mathcal{F})$.