Erdős Problem 146: Failure of the 2-Degenerate Extremal BoundResearch Paper
A graph $H$ is **$r$-degenerate** if every nonempty subgraph of $H$ has a vertex of degree at most $r$. Erdős conjectured — this is Erdős problem #146 — that every fixed bipartite $r$-degenerate graph $H$ satisfies
$$\mathrm{ex}(n, H) = O\!\left(n^{2-1/r}\right).$$
The conjecture was known in several cases: when one bipartition class has maximum degree at most $r$, for $r$-degenerate blow-ups of trees, and, for $r = 2$, for grids and certain critical 2-degenerate graphs. The best general bound was the weaker $\mathrm{ex}(n,H) = O(n^{2-1/(4r)})$ of Alon, Krivelevich and Sudakov.
This mission carries a complete Lean 4 formalisation **refuting** it at $r = 2$.
**Theorem.** There exist a fixed connected bipartite 2-degenerate graph $H$ and constants $c, \varepsilon > 0$ such that
$$\mathrm{ex}(n, H) \ \ge\ c\,n^{3/2 + \varepsilon}$$
for all sufficiently large $n$. Since the conjectured bound at $r = 2$ is $O(n^{3/2})$, the excess is polynomial rather than constant, so the conjecture fails outright. A related conjecture of Erdős (problem #113) asserts that a bipartite graph is 2-degenerate if and only if $\mathrm{ex}(n,H) = O(n^{3/2})$; Janzer had already disproved the reverse implication, and this result refutes the forward one.
**The construction.** The counterexample $H$ is built in layers: starting from a layer $V_0$ of size $L_0$, each subsequent layer is $V_i = \binom{V_{i-1}}{2}$, and every vertex $\{a,b\} \in V_i$ is joined to its two parents $a, b \in V_{i-1}$. The result is connected, bipartite and 2-degenerate by construction, and is related to the complete degenerate graphs of Grzesik, Janzer and Nagy.
The lower bound comes from a **sampled Hamming-ball graph**. With $U = \{0,1\}^m$, two disjoint copies $U_L, U_R$ are joined whenever their Hamming distance is at most $k = \lfloor \tau m\rfloor$, and each vertex is retained independently with probability $p = 2^{-\beta m}$. The two parameters are governed by the thresholds
$$A(\tau) = \kappa + \tau\log_2 3, \qquad C(\tau) = 2h(\tau) - 1,$$
and the construction needs a sampling exponent with $A(\tau) < \beta < C(\tau)$. The lower threshold controls exclusion of the layered graph; the upper one controls whether the sampled host has more than $n^{3/2}$ edges.
Exclusion runs on a conditional-entropy functional $E(u,z) = \frac{1}{m}\sum_j H(Z_j \mid X_j, Y_j)$ over parent and child arrays. An array of conditional entropy $E$ has at most $2^{mME + O(m\log_2 M)}$ realisations, while requiring its $M = \binom{L}{2}$ children to survive sampling costs $2^{-\beta mM}$ — which dominates the $2^{mL}$ possible parent arrays whenever $E < \beta$. An embedding of $H$ would therefore have to raise a bounded entropy potential by a fixed amount at each layer, which is impossible after enough layers. A second-moment argument shows the sampled graph still has $\Omega(n^{3/2+\varepsilon})$ edges, and padding extends the construction to every sufficiently large order.
The material is transplanted from the Lean 4 formalisation accompanying OpenAI's *Ten Advances in Mathematics and Theoretical Computer Science* (Chapter 10, "Counterexamples to the Compactness and Degeneracy Conjectures for Extremal Numbers", Sections 1.2 and 5–8), and re-verified in this environment: every node is proved from `[propext, Classical.choice, Quot.sound]` alone, and each staged statement's elaborated type was checked to be identical to the original declaration's. The mission is offered as a curated, closed campaign whose definitions and lemmas — binary entropy and the pair kernel, the layered construction, the Hamming-ball host and its retention measure — are reusable foundations for further work in extremal graph theory.
This is the companion result to Erdős problem #180, the Erdős–Simonovits compactness conjecture, which is formalised in the same source chapter and published as a separate mission.
Erdős Problem 180: the Erdős–Simonovits Compactness ConjectureResearch Paper
Erdős and Simonovits conjectured that forbidding a finite family of graphs cannot reduce the extremal number by more than a constant factor compared with forbidding one of its members: for every finite nonempty family $\mathcal{F}$ whose members all contain a cycle, there should be some $F \in \mathcal{F}$ and $C>0$ with $\mathrm{ex}(n,F) \le C\,\mathrm{ex}(n,\mathcal{F})$ for all large $n$. The cycle hypothesis is essential — the folklore family $\{K_{1,2}, 2K_2\}$ already defeats the original formulation — and the corrected conjecture is Erdős problem #180.
This mission carries a complete Lean 4 formalisation refuting it, and refuting it quantitatively: there is a finite family $\mathcal{F}$ of **connected bipartite** graphs, each containing a cycle, with
$$\mathrm{ex}(n,\mathcal{F}) = O\!\left(n^{4/3-1/48}\right) \qquad\text{while}\qquad \mathrm{ex}(n,F) = \Omega\!\left(n^{4/3}\right) \ \ (F \in \mathcal{F}).$$
The two bounds are separated by a polynomial factor $n^{1/48}$, so no member can dominate the family up to any constant. The family is $\mathcal{F} = \{C_4, C_6\} \cup \mathcal{J} \cup \mathcal{K}$, where $\mathcal{J}$ and $\mathcal{K}$ are the admissible quotients of two properly $2$-coloured templates built from the subdivisions of $K_{3,2}$ and $K_{3,3}$. The upper bound comes from counting short paths in an $\mathcal{F}$-free graph: excluding $\mathcal{J}$ bounds the number of vertices that fail to be centres of a subdivided $K_{3,3}$, and excluding $\mathcal{K}$ forces those vertices to form a vertex cover. The lower bound comes from incidence graphs of symplectic generalized quadrangles $W(q)$, with the characteristic of the underlying field chosen to suit the forbidden member — even $q$ for $\mathcal{J}$, odd $q$ for $\mathcal{K}$ — which is exactly the freedom a family bound does not have.
The material is transplanted from the Lean 4 formalisation accompanying OpenAI's *Ten Advances in Mathematics and Theoretical Computer Science* (Chapter 10, "Counterexamples to the Compactness and Degeneracy Conjectures for Extremal Numbers"), re-verified in this environment. Every node is proved; the mission is offered as a curated, closed campaign whose definitions and lemmas are reusable foundations for further work in extremal graph theory.
Erdős Problem 183: Multicolour Triangle Ramsey NumbersResearch Paper
How fast do multicolour Ramsey numbers grow? Write $R_k$ for the least $n$ such that every colouring of the edges of $K_n$ with $k$ colours contains a monochromatic triangle. The classical bounds, essentially unimproved for decades, place $R_k$ between $c^k$ and $e\cdot k!$, and Erdős asked repeatedly whether the truth is closer to the exponential lower end — his Problem 183 asks whether $R_k^{1/k}\to\infty$, i.e. whether the growth is genuinely superexponential.
This mission carries a complete Lean 4 formalisation resolving that question in the affirmative, with an explicit bound: $R_k \ge \left(\tfrac{1}{6e^{38}}\,k^{1/3}/\log k\right)^{k}$ for all sufficiently large $k$, from which $R_k^{1/k}\to\infty$ follows, together with the matching two-sided estimate $\log R_k = \Theta(k\log k)$ pinning the sharp coefficients. The argument is constructive: it builds triangle-free colourings by a recursive palette construction whose colour count grows fast enough to beat every exponential.
The material is transplanted from the Lean 4 formalisation accompanying OpenAI's *Ten Advances in Mathematics and Theoretical Computer Science*, re-verified in this environment. Every node is proved — the mission is offered as a curated, closed campaign whose milestones map the attack path and whose lemmas are reusable foundations for further work on multicolour Ramsey theory.
Erdős Problem 788: Exponent One-Half and Explicit BoundsResearch Paper
Erdős Problem 788 asks how large a set can always be retained when prescribed distinct pair-sums are forbidden. This mission formalizes the repository’s strengthened version of Theorem 1.1: an explicit lower bound valid for every $n\ge 3$, an eventual quantitative upper bound, the conclusion $f(n)=n^{1/2+o(1)}$, and the exact affirmative answer to the original upper-bound question.
Bandit feedback is only one point on a spectrum: a learner might see more than its own loss (full information) or less (a spam filter never learns what happened to mail it deleted). Chapter 37 of Lattimore–Szepesvári studies finite adversarial games $G = (\mathcal{L}, \Phi)$ where the loss matrix and the feedback matrix are decoupled. The goal theorem is the celebrated classification theorem: every finite partial-monitoring game has minimax regret exactly $0$, $\Theta(\sqrt{n})$, $\Theta(n^{2/3})$ or $\Omega(n)$ — determined by two purely combinatorial conditions, global and local observability, on the game's neighbourhood structure. A single geometric dichotomy thus governs the price of information in every online decision problem with finite actions and feedback.
Bandit Algorithms XVI: Markov Decision Processes and UCRL2Textbook
The final step from bandits to reinforcement learning: actions now change the state of the world. Chapter 38 of Lattimore–Szepesvári studies online learning in an unknown Markov decision process with $S$ states, $A$ actions and rewards in $[0,1]$. The optimism principle of Mission II scales up: UCRL2 maintains confidence sets over transition kernels, solves an *extended* MDP by extended value iteration, and recomputes only when a state-action count doubles. The goal theorem: with probability $1-\delta$, $\hat R_n < C\,D(M)\,S\sqrt{An\log(nSA/\delta)}$, where $D(M)$ is the diameter of the MDP — sublinear regret with no prior knowledge of the dynamics. The matching lower bound $\mathbb{E}[\hat R_n] \ge C'\sqrt{DSAn}$ brackets the true complexity of tabular reinforcement learning up to $\sqrt{DS}$.
Bandit Algorithms XIV: Bayesian Bandits, the Gittins Index and Thompson SamplingTextbook
The oldest bandit algorithm (Thompson, 1933) is also the most modern: sample a parameter from the posterior and act greedily. Chapters 34–36 of Lattimore–Szepesvári develop the Bayesian view in two crowning results. The Gittins index theorem: for infinite-horizon discounted Markov bandits, the seemingly intractable dynamic program is solved *exactly* by an index policy — each arm gets a retirement-value index computable arm-by-arm, and playing the largest index is Bayesian optimal. And the frequentist analysis of Thompson sampling — the goal theorem: with Gaussian posteriors, Thompson sampling on 1-subgaussian bandits achieves $\lim_{n\to\infty} R_n/\log n = \sum_{i:\Delta_i>0} 2/\Delta_i$, exactly asymptotically optimal, alongside the minimax-grade $R_n \le C\sqrt{kn\log n}$. Together they explain why posterior sampling is both principled and practically dominant.
Bandit Algorithms XIII: Pure Exploration and Best-Arm IdentificationTextbook
Sometimes reward during learning is irrelevant — a pharmaceutical company running phase-II trials only cares about identifying the best treatment, as quickly and as reliably as possible. Chapter 33 of Lattimore–Szepesvári formalizes fixed-confidence best-arm identification: a policy together with a stopping time $\tau$ and a recommendation must be *sound* (wrong with probability at most $\delta$) while minimizing $\mathbb{E}[\tau]$. The information-theoretic complexity is $c^*(\nu)^{-1} = \sup_{\alpha\in\mathcal{P}_{k-1}} \inf_{\nu'\in\mathcal{E}_{alt}(\nu)} \sum_i \alpha_i D(\nu_i, \nu_i')$: every sound strategy needs $\mathbb{E}[\tau] \ge c^*(\nu)\log\frac{1}{4\delta}$, and the Track-and-Stop algorithm — the goal theorem — achieves $\lim_{\delta\to 0} \mathbb{E}[\tau]/\log(1/\delta) = c^*(\nu)$ exactly. The mission also covers the fixed-budget counterpart, sequential halving.
Bandit Algorithms XII: Follow-the-Regularised-Leader and Mirror DescentTextbook
Beneath Exp3, Exp4 and their relatives lies one algorithm: minimize past losses plus a convex regularizer. Chapters 26–28 of Lattimore–Szepesvári develop this unifying view. For a Legendre potential $F$ with Bregman divergence $D_F$, both mirror descent and follow-the-regularised-leader satisfy the master bound $R_n(a) \le \frac{F(a) - F(a_1)}{\eta} + \frac{1}{\eta}\sum_t D_F(a_t, \tilde a_{t+1})$; the negentropy potential on the simplex recovers Exp3 exactly. The goal theorem is the payoff for adversarial *linear* bandits: FTRL on the unit ball with the self-concordant-flavoured potential $F(a) = -\log(1-\|a\|) - \|a\|$ achieves $R_n \le 2\sqrt{3nd\log n}$ — improving the $\sqrt{d}$ factor over the Exp3-style approach of Chapter 27 and matching the $\Omega(d\sqrt{n})$ lower bound of Mission XI up to logarithms.
Bandit Algorithms XI: Lower Bounds for Stochastic Linear BanditsTextbook
Is the $d\sqrt{n}$ regret of LinUCB (Mission X) an artifact of the algorithm or a law of nature? Chapters 24–25 of Lattimore–Szepesvári prove it is essentially unimprovable. On the unit ball there is a parameter $\theta$ with $\|\theta\|_2^2 = d^2/(48n)$ forcing $R_n \ge \frac{d\sqrt{n}}{16\sqrt{3}}$ — the goal theorem — and the hypercube gives the same $\Omega(d\sqrt{n})$ rate. The asymptotic chapter is more striking still: for fixed finite action sets, the instance-optimal constant $c(\mathcal{A},\theta)$ is characterized by an allocation program, and *optimism itself is provably suboptimal* — LinUCB and Thompson sampling cannot achieve it, because exploration must sometimes deliberately play actions optimism would never touch. These lower bounds define the targets for the entire linear-bandit literature.
Bandit Algorithms X: Stochastic Linear Bandits and LinUCBTextbook
When actions are feature vectors and the mean reward is linear — $X_t = \langle \theta_*, A_t\rangle + \eta_t$ — a bandit can generalize across arms: pulling one arm reveals information about all of them. Chapter 19 of Lattimore–Szepesvári carries the optimism principle into this setting: LinUCB (a.k.a. OFUL) plays the action maximizing $\max_{\theta\in\mathcal{C}_t}\langle\theta, a\rangle$ over the confidence ellipsoid $\mathcal{C}_t$ of Mission IX. The goal theorem: with probability $1-\delta$, $\hat R_n \le \sqrt{8n\beta_n \log\frac{\det V_n}{\det V_0}} \le \sqrt{8dn\beta_n\log\frac{d\lambda + nL^2}{d\lambda}}$ — regret $\tilde O(d\sqrt{n})$ independent of the number of actions. The combinatorial engine is the elliptical potential lemma, bounding how many times adaptively chosen directions can be surprising. Chapter 22's phased elimination with G-optimal design (Mission IX) sharpens this to $\tilde O(\sqrt{dn\log k})$ for finite action sets.
Bandit Algorithms IX: Self-Normalized Concentration and Optimal DesignTextbook
Least-squares estimation from *adaptively* collected data is the statistical heart of linear bandits: the actions $A_t$ depend on past noise, so classical fixed-design theory does not apply. Chapter 20 of Lattimore–Szepesvári resolves this with the method of mixtures: the process $M_t(x) = \exp(\langle x, S_t\rangle - \frac{1}{2}\|x\|^2_{V_t(\lambda)})$ is a supermartingale, and integrating over a Gaussian mixture yields the self-normalized bound — the goal theorem — $\mathbb{P}\big(\exists t : \|S_t\|^2_{V_t(\lambda)^{-1}} \ge 2\log\frac{1}{\delta} + \log\frac{\det V_t(\lambda)}{\lambda^d}\big) \le \delta$, valid uniformly over all times. The resulting confidence ellipsoids for the regularized least-squares estimator (Abbasi-Yadkori et al.) calibrate every algorithm of Mission X. The mission also formalizes the Kiefer–Wolfowitz theorem of Chapter 21: G-optimal and D-optimal experimental designs coincide, with optimal value exactly $d$ — the classical equivalence theorem of optimal design theory.
Bandit Algorithms VIII: Contextual Bandits and Exp4Textbook
Real decisions come with context: a news site chooses an article *for a particular user*. Competing with the single best arm is then meaningless; the right benchmark is the best mapping from contexts to arms, or more generally the best of $M$ expert policies. Chapter 18 of Lattimore–Szepesvári formalizes this via Exp4 — exponential weighting over experts, fed by the importance-weighted estimator of Mission V. The goal theorem: with learning rate $\eta = \sqrt{2\log(M)/(nk)}$, Exp4 satisfies $R_n \le \sqrt{2nk\log M}$ against the best of $M$ experts. Since $M$ enters only logarithmically, the learner can compete with exponentially large policy classes — the conceptual gateway from bandits to reinforcement learning with function approximation.
A continuous piecewise linear (CPWL) function is one assembled from finitely many flat pieces glued along flat seams. Every ReLU network computes such a function, and every such function is computed by some ReLU network. Questions about how deep a network must be are therefore questions about the internal structure of CPWL functions.
In 1993 Breiman built such functions from hinges: maxima of two affine maps. Sums of hinges approximate anything, but from two dimensions up they fail to represent most CPWL functions exactly. Wang and Sun (2005) widened the maxima, proving that every CPWL function on ℝⁿ is a signed sum of maxima of at most n+1 affine maps. Twenty years on it remains the workhorse structural fact, reducing any question about a network to a question about a single max gate and underpinning every known upper bound on the depth of exact representation.
That includes the newest one: at STOC 2026, Bakaev et al disproved the short standing conjecture that ⌈log₂(n+1)⌉ hidden layers are necessary, showing ⌈log₃(n−1)⌉+1 suffice. In this mission we deliver a machine-checked proof of the Wang and Sun theorem so future formalizations of network expressivity can invoke it rather than reprove it. Note that we take as given the lattice representation of Tarela and Martínez, independently proved by Ovchinnikov, which writes any CPWL function as a max of mins of its affine pieces. That is the one external ingredient the argument consumes, and our definition of CPWL builds it in.
Bandit Algorithms IV: Bernoulli Bandits and KL-UCBTextbook
When rewards are binary — a click or no click, a cure or no cure — the subgaussian machinery of Missions I–III is not tight: the variance of a Bernoulli arm degrades near the boundary of $[0,1]$, and the correct exponential rate is governed by the binary relative entropy $d(p,q) = p\log\frac{p}{q} + (1-p)\log\frac{1-p}{1-q}$ rather than a squared distance. Chapter 10 of Lattimore–Szepesvári develops Chernoff's tail bound in its information-theoretic form and the KL-UCB algorithm, whose upper confidence bounds are level sets of $d$. The goal theorem shows KL-UCB attains $$\limsup_{n\to\infty} R_n/\log n = \sum_{i:\Delta_i>0} \Delta_i/d(\mu_i, \mu^*)$$ — asymptotic optimality with exactly the constant demanded by the lower bound of Mission VII, strictly improving subgaussian UCB on every Bernoulli instance.
Randomness appears to enlarge efficient computation, but the Sipser–Gács–Lautemann theorem places every bounded-error probabilistic polynomial-time language at the second level of the polynomial hierarchy, giving one of complexity theory’s foundational limits on the power of randomization.
Bandit Algorithms VII: Lower Bounds for Finite-Armed BanditsTextbook
How well can any algorithm possibly do? Chapters 13–17 of Lattimore–Szepesvári answer with three matching impossibility results. The divergence decomposition identifies the information a policy collects: $D(\mathbb{P}_{\nu\pi}, \mathbb{P}_{\nu'\pi}) = \sum_i \mathbb{E}[T_i(n)] D(P_i, P_i')$. Feeding it into the Bretagnolle–Huber inequality of Mission VI yields the goal theorem — the minimax lower bound $R_n \ge \frac{1}{27}\sqrt{(k-1)n}$ over Gaussian bandits, showing MOSS (Mission III) is optimal up to a constant. The same machinery gives the instance-dependent bound of Lai–Robbins type: every consistent policy suffers $\liminf_n R_n/\log n \ge \sum_{i:\Delta_i>0} \Delta_i / d_{\inf}(P_i, \mu^*, \mathcal{M}_i)$, certifying the asymptotic optimality of the UCB of Mission III and KL-UCB of Mission IV, and a high-probability lower bound showing the Exp3-IX guarantees of Mission V cannot be improved.
Bandit Algorithms III: Asymptotic and Minimax Optimality of UCBTextbook
The basic UCB regret bound of Mission II is logarithmic but not tight: its leading constant $16/\Delta_i$ is eight times the information-theoretic limit, and its worst-case rate carries a spurious $\sqrt{\log n}$. Chapters 8–9 of Lattimore–Szepesvári close both gaps. A refined confidence schedule $f(t) = 1 + t\log^2 t$ yields the asymptotically optimal $\limsup_{n\to\infty} R_n/\log n \le \sum_{i:\Delta_i>0} 2/\Delta_i$ — exactly matching the instance-dependent lower bound of Mission VII for Gaussian noise. The MOSS index $\hat\mu_i + \sqrt{\tfrac{4}{T_i}\log^+\!\big(\tfrac{n}{k T_i}\big)}$ achieves minimax regret $R_n \le 39\sqrt{kn} + \sum_i \Delta_i$, matching the $\Omega(\sqrt{kn})$ lower bound up to a constant. These two theorems are the gold standard for finite-armed stochastic bandits.
Every lower bound in bandit theory rests on one question: how hard is it to tell two probability measures apart from a sample? The answer is quantified by the relative entropy $D(P,Q)$, and the sharpest elementary tool is the Bretagnolle–Huber inequality: for any event $A$, $P(A) + Q(A^c) \ge \frac{1}{2}\exp(-D(P,Q))$ — no test can distinguish $P$ from $Q$ with total error probability below $\frac{1}{2}e^{-D(P,Q)}$. This mission formalizes Chapter 14 of Lattimore–Szepesvári: the Bretagnolle–Huber inequality (the goal theorem, proved via Le Cam's inequality $\int p \wedge q \ge \frac{1}{2}(\int\sqrt{pq})^2$), Pinsker's inequality $\delta(P,Q) \le \sqrt{D(P,Q)/2}$, and the closed-form divergences between Gaussians and Bernoullis. These half-page inequalities power every impossibility result in Missions VII, XI and beyond.
Bandit Algorithms V: Adversarial Bandits and Exp3Textbook
What if the rewards are not random at all, but chosen by an adversary who knows your algorithm? Remarkably, a randomized learner can still compete with the best fixed arm in hindsight. Chapters 11–12 of Lattimore–Szepesvári develop the adversarial $k$-armed bandit: rewards $x_{ti} \in [0,1]$ are an arbitrary fixed matrix, the learner samples $A_t \sim P_t$, and regret is measured against $\max_i \sum_t x_{ti}$. The exponential-weights algorithm Exp3, fed by importance-weighted loss estimates $\hat X_{ti} = 1 - \mathbb{1}\{A_t = i\}(1 - X_t)/P_{ti}$, achieves $R_n \le \sqrt{2nk\log k}$ — the goal theorem. The companion Exp3-IX, which deliberately biases its estimator, upgrades this to a bound holding with high probability rather than only in expectation. These results are the foundation of all adversarial online learning with partial feedback.
Bandit Algorithms II: Stochastic Bandits and the UCB AlgorithmTextbook
A learner repeatedly chooses one of $k$ slot machines, observes only the reward of the chosen arm, and wants to earn almost as much as the best arm in hindsight. This is the stochastic multi-armed bandit, the canonical model of the exploration–exploitation dilemma. This mission formalizes the model (environments, policies, regret, and the regret decomposition $R_n = \sum_i \Delta_i\,\mathbb{E}[T_i(n)]$) and the two classical algorithms of Chapters 6–7 of Lattimore–Szepesvári: Explore-Then-Commit and the Upper Confidence Bound algorithm built on the optimism principle. The goal theorem is the instance-dependent UCB regret bound $R_n \le 3\sum_i \Delta_i + \sum_{i:\Delta_i>0} 16\log(n)/\Delta_i$ — logarithmic regret with explicit constants, the single most cited result of bandit theory — together with its distribution-free companion $R_n \le 8\sqrt{nk\log n} + 3\sum_i \Delta_i$.
Bandit Algorithms I: Concentration of MeasureTextbook
How quickly does the empirical mean of independent random variables concentrate around the true mean? This question is the analytic engine of the entire theory of stochastic bandits: every optimistic algorithm (Explore-Then-Commit, UCB and its relatives) is calibrated by a tail bound on the sample mean. This mission formalizes the subgaussian framework of Chapter 5 of Lattimore–Szepesvári's *Bandit Algorithms*: a random variable $X$ is $\sigma$-subgaussian when $\mathbb{E}[e^{\lambda X}] \le e^{\lambda^2\sigma^2/2}$ for all $\lambda$, and the Cramér–Chernoff method converts this moment-generating-function control into the exponential tail $\mathbb{P}(X \ge \varepsilon) \le e^{-\varepsilon^2/(2\sigma^2)}$. The goal theorem is the Hoeffding-type bound: the sample mean of $n$ independent $\sigma$-subgaussian deviations exceeds the true mean by $\varepsilon$ with probability at most $\exp(-n\varepsilon^2/(2\sigma^2))$, together with its confidence form $\mathbb{P}\big(\hat\mu + \sqrt{2\sigma^2\log(1/\delta)/n} \le \mu\big) \le \delta$ — the exact bound every UCB index is built from. These few lines of analysis are cited by every regret bound in the series.
Markov Entanglement: Decomposition Error via Agent-wise TV DistanceResearch Paper
Multi-agent reinforcement learning approximates a global value function by summing per-agent local value functions learned independently — a trick that works surprisingly well in practice (ride-hailing dispatch, restless bandits) but had no general theoretical justification. Chen and Peng (arXiv:2506.02385) explain why: they define a *Markov entanglement* measure for the joint transition dynamics of a multi-agent MDP, directly analogous to quantum entanglement of a two-party state, and show it controls exactly how much error this value-decomposition trick incurs. This mission formalizes their sharpest quantitative bound (Theorem 4): the error of decomposing the global Q-function into per-agent local Q-functions is controlled, entrywise, by the agent-wise total-variation measure of Markov entanglement.
Single-Server Queueing Convergence via Forward CouplingTextbook
Formalize sample-path stability for a continuous-time, unit-rate, infinite-buffer single-server queue. Starting from cumulative arriving service work, define the reflected transient workload, the workload constructed from the infinite past, long-run offered load, and two-time stationarity. Prove that subcritical load forces finite-time coupling and consequently that every finite initial workload converges in its two-time finite-dimensional distributions to the stationary workload law.
Formalize sample-path Little's Law for deterministic continuous-time queueing trajectories, decomposed into area, sojourn, arrival-rate, boundary, and squeeze lemmas.