We improve the upper bound for diagonal Ramsey numbers to R(k+1,k+1)≤exp(−c(logk)2)2kk k≥3. To do so, we build on a quasirandomness and induction framework introduced by Thomason extended Conlon, demonstrating optimal "effective quasirandomness" results about convergence of graphs. This optimality represents natural barrier improvement.

Abstract In 1981, Karp and Sipser proved a law of large numbers for the matching number sparse Erdős–Rényi random graph, in an influential paper pioneering so‐called differential equation method analysis graph processes. Strengthening this classical result, answering question Aronson, Frieze Pittel, we prove central limit theorem same setting: fluctuations are asymptotically Gaussian. Our new contribution is to subcritical critical regimes, according celebrated algorithmic phase transition...

Although class B audio-frequeny amplifiers have been analyzed by many previous investigators, the effect of leakage inductance output transformer or choke in producing quasi transients, i.e., exponential terms which recur periodically, has passed unnoticed. This paper gives equations for determining these transients wave forms plate voltage, current, and when amplifier reached a permanent state. The theoretical relations are derived from fundamental involving tube characteristic, is assumed...

The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ a finite abelian group order $N$, $A\subseteq G^k$ with $|A|\ge \alpha N^k$. If $M_2$, $M_1-M_2$, $M_1+M_2$ are automorphisms $G^k$, is it true that there exists popular difference $d \in G^k\setminus \{0\}$ such \[ \#\{x G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d A\} \ge (\alpha ^4-o(1))N^k?\] We show this false in...

Very sparse random graphs are known to typically be singular (i.e., have adjacency matrix) due the presence of "low-degree dependencies" such as isolated vertices and pairs degree 1 with same neighborhood. We prove that these kinds dependencies in some sense only causes singularity: for constants k≥3 λ>0, an Erdős–Rényi graph G∼G(n,λ∕n) n edge probability λ∕n has property its k-core (its largest subgraph minimum at least k) is nonsingular. This resolves a conjecture Vu from 2014...

.We give an online algorithm that with high probability computes a \( (\frac{e}{e-1} + o(1) )\Delta\) edge coloring on graph \(G\) maximum degree \(\Delta = \omega (\log n)\) under arrivals against oblivious adversaries, making first progress the conjecture of Bar-Noy, Motwani, and Naor in this general setting. Our is based reducing to matching problem locally treelike graphs, then applying tree recurrence approach for arguing correlation decay.Keywordsedge coloringonline algorithmstree...

Resolving a conjecture of Füredi from 1988, we prove that with high probability, the random graph <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G left-parenthesis n comma 1 slash 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo>...

Abstract Conditional on the extended Riemann hypothesis, we show that with high probability, characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses question raised by Eberhard in recent work. The main innovation our work establishing sharp estimates regarding rank distribution -matrices over $\mathbb{F}_p$ for primes $2 &lt; p \leq \exp(O(n^{1/4}))$ . Previously, such were available only $p = o(n^{1/8})$ At heart proof way to combine multiple...

In this note, we study large deviations of the number N $\mathbf {N}$ intercalates ( 2 × $2\times 2$ combinatorial subsquares which are themselves Latin squares) in a random n $n\,{\times}\, n$ square. particular, for constant δ > 0 $\delta >0$ prove that exp − O log ) ⩽ Pr 1 / 4 Ω $\exp (-O(n^{2}\log n))\leqslant \Pr (\mathbf {N}\leqslant (1-\delta )n^{2}/4)\leqslant \exp (-\Omega (n^{2}))$ and 3 ⩾ + (-O(n^{4/3}(\log n)))\leqslant {N}\geqslant (1+\delta (n^{4/3}(\log n)^{2/3}))$ . As...

We give an FPTAS for computing the number of matchings size k in a graph G maximum degree Δ on n vertices, all ≤ (1−δ)m*(G), where δ>0 is fixed and m*(G) matching G, independent sets (1−δ) αc(Δ) n, NP-hardness threshold this problem. also provide quasi-linear time randomized algorithms to approximately sample from uniform distribution (1−δ)m*(G) (1−δ)αc(Δ)n.

In this work, we analyze dimension reduction algorithms based on the Kac walk and discrete variants. (1) For n points in Rd, design an optimal Johnson–Lindenstrauss (JL) transform which can be applied to any vector time O(dlogd) for essentially same restriction as best-known transforms due Ailon Liberty, Bamberger Krahmer. Our algorithm is memory-optimal, outperforms existing regimes when sufficiently large distortion parameter small. particular, confirms a conjecture of Chazelle, Oliveira,...

We derive an explicit distribution for the threshold sequence of symmetric binary perceptron with Gaussian disorder, proving that critical window is constant width.

Let A $A$ be an n × $n\times n$ real matrix, and let M $M$ random matrix whose entries are independent identically distributed sub-Gaussian variables with mean 0 variance 1. We make two contributions to the study of s ( + ) $s_n(A+M)$ , smallest singular value $A+M$ . (1) show that for all ε ⩾ $\epsilon \geqslant 0$ P [ ⩽ ] = O 2 e − Ω \begin{equation*} \mathbb {P}[s_n(A M) \leqslant \epsilon O(\epsilon \sqrt {n}) 2e^{-\Omega (n)}, \end{equation*} provided only has $\Omega (n)$ values which...

Abstract May the triforce be 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that minimum density in a of edge δ is 4– o (1) but not O ( 4 ). Let M ) maximum number such following holds: for every ∊ &gt; 0 and $G = {\mathbb{F}}_2^n$ n sufficiently large, if A ⊆ G × ≥ | 2 , then there exists nonzero “popular difference” d ∈ “corners” x y ), + at least )–∊)| . As corollary via recent result Mandache, we conclude ω On other hand, &lt; 1/2 large N [ ] 3 δN ≠ 0,...