Let left bracket q right bracket equals StartSet 0 comma 1 comma ellipsis comma q minus 1 EndSet [ q ] = { 0 , 1 , … , q − 1 } $[q] = \{0,1,\ldots ,q-1\}$ , let normal upper Delta left bracket q right bracket Δ [ q ] $\Delta [q]$ denote the simplex of probability measures on left bracket q right bracket [ q ] $[q]$ , and let gamma γ $\gamma$ denote the Lebesgue measure normalized on normal upper Delta left bracket q right bracket Δ [ q ] $\Delta [q]$ . We prove that for any symmetric monotone function f colon left bracket q right bracket Superscript n Baseline right arrow left bracket q right bracket f : [ q ] n → [ q ] ${\kern1pt}f \colon{\kern-1pt} [q]^n \to [q]$ and any a element of left bracket q right bracket a ∈ [ q ] $a \in [q]$ , we have
Let Omega(1), ... , Omega(m) be probability spaces, let Omega =Omega(1) & times; & centerdot;& centerdot;& centerdot; & times; Omega(m) be their product and let A(1), ... , A(n) subset of Omega be events. Suppose that each event A(i) depends on ri coordinates of a point x is an element of Q, x = (xi(1), ... ,xi(m)), and that for each event A(i) there are A(i) other events A(j) that depend on some of the coordinates that A(i) depends on. Let A = max{5, Ai: i= 1, ... , n} and let & micro;(i) = min{r(i), Delta(i) + 1} for i = 1, ... , n. We prove that if P(A(i)) < (3 Delta)(-3 & micro;i) for all i, then for any 0 < E < 1, the probability P boolean AND(n)(i=1) (A) over bar (-)(i) of the intersection of the complements of all Ai can be computed within relative error is an element of in polynomial time from the probabilities P (A(i1) boolean AND ... boolean AND A(ik)) of k-wise intersections of the events A(i) for k = e(O(Delta)) ln (n/is an element of).
Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$ . Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$ . In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$ . This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$ .
We prove that several natural graph classes have tree-decompositions with minimum width such that each bag has bounded treewidth. For example, every planar graph has a tree-decomposition with minimum width such that each bag has treewidth at most 3. This treewidth bound is best possible. More generally, every graph of Euler genus g has a tree-decomposition with minimum width such that each bag has treewidth in O(g). This treewidth bound is best possible. Most generally, every K_p-minor-free graph has a tree-decomposition with minimum width such that each bag has treewidth at most some polynomial function f(p). In such results, the assumption of an excluded minor is justified, since we show that analogous results do not hold for the class of 1-planar graphs, which is one of the simplest non-minor-closed monotone classes. In fact, we show that 1-planar graphs do not have tree-decompositions with width within an additive constant of optimal, and with bags of bounded treewidth. On the other hand, we show that 1-planar n-vertex graphs have tree-decompositions with width O(√(n)) (which is the asymptotically tight bound) and with bounded treewidth bags. Moreover, this result holds in the more general setting of bounded layered treewidth, where the union of a bounded number of bags has bounded treewidth.
We study off-diagonal Ramsey numbers $r(H, K_n<^>{(k)})$ of $k$ -uniform hypergraphs, where $H$ is a fixed linear $k$ -uniform hypergraph and $K_n<^>{(k)}$ is complete on $n$ vertices. Recently, Conlon, Fox, Gunby, He, Mubayi, Suk, and Verstra & euml;te disproved the folklore conjecture that $r(H, K_n<^>{(3)})$ always grows polynomially in $n$ . In this paper, we show that much larger growth rates are possible in higher uniformity. In uniformity $k\ge 4$ , we prove that for any constant $C\gt 0$ , there exists a linear $k$ -uniform hypergraph $H$ for which \begin{equation*} r(H,K_n<^>{(k)}) \geq { extrm {twr}}_{k-2}(2<^>{(\log n)<^>C}). \end{equation*}
We prove that for any k >= 3 for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random k-SAT formulas is given by the 'replica symmetric solution' predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. 76 (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any k >= 3, we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)]. The improvement is significant particularly for small k.
We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.
Confirming a conjecture of Erd & odblac;s on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lov & aacute;sz proved that in any $q$ -colouring of the edges of the complete $r$ -uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac {n+q-1}{r+q-1}\rfloor$ . In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$ , we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$ -colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon-Frankl-Lov & aacute;sz theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erd & odblac;s matching conjecture.
Given a collection $\mathcal{D} =\{D_1,D_2,\ldots ,D_m\}$ of digraphs on the common vertex set $V$ , an $m$ -edge digraph $H$ with vertices in $V$ is transversal in $\mathcal{D}$ if there exists a bijection $\varphi \,:\,E(H)\rightarrow [m]$ such that $e \in E(D_{\varphi (e)})$ for all $e\in E(H)$ . Ghouila-Houri proved that any $n$ -vertex digraph with minimum semi-degree at least $\frac {n}{2}$ contains a directed Hamilton cycle. In this paper, we provide a transversal generalisation of Ghouila-Houri's theorem, thereby solving a problem proposed by Chakraborti, Kim, Lee, and Seo. Our proof utilises the absorption method for transversals, the regularity method for digraph collections, as well as the transversal blow-up lemma and the related machinery. As an application, when $n$ is sufficiently large, our result implies the transversal version of Dirac's theorem, which was proved by Joos and Kim.
A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$ -density contains a logarithmic blowup of $K_r$ . In this paper, we explore variants of Nikiforov's result in the following form. Given $r,t\in \mathbb{N}$ , when a positive $K_{r}$ -density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$ ? Our results include:For an $n$ -vertex ordered graph $G$ with no induced monotone path $P_{6}$ , if its complement $\overline {G}$ has positive triangle density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$ . This strengthens a recent result of Pach and Tomon. For general $k$ , let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$ -vertex ordered graph $G$ with no induced monotone $P_{2k}$ , if $\overline {G}$ has positive $K_r$ -density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$ . Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2. Any incomparability graph with positive $K_{r}$ -density contains a blowup of $K_r$ of size $\Omega ({n \over {\log n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$ -comparability graph with positive $K_{(2h-2)<^>{r}+1}$ -density contains a blowup of $K_h$ of size $\Omega (n)$ , where the constant $(2h-2)<^>{r}+1$ is optimal.The ${n \over {\log n}}$ size of the blowups in all our results are optimal up to a constant factor.
In this article, we study a non-uniform distribution on permutations biased by their number of records that we call record-biased permutations. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterise. This model was introduced in our earlier work [3], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.
A trace of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied trace reconstruction problem asks how many traces are required to reconstruct an unknown binary sequence with high probability. In this paper, we study the multidimensional version of this problem for matrices and hypermatrices, where a trace is generated by deleting each row/column of the matrix or each slice of the hypermatrix independently with a constant probability. Previously, Krishnamurthy, Mazumdar, McGregor and Pal showed that $\exp (\widetilde {O}(n<^>{d/(d+2)}))$ traces suffice to reconstruct any unknown $n\times n$ matrix (for $d=2$ ) and any unknown $n<^>{\times d}$ hypermatrix. By developing a dimension reduction procedure and establishing a multivariate version of the Littlewood-type result that lower bounds sparse complex polynomials around $1$ , we improve this upper bound by showing that $\exp (\widetilde {O}(n<^>{3/7}))$ traces suffice to reconstruct any unknown $n\times n$ matrix, and $\exp (\widetilde {O}(n<^>{3/5}))$ traces suffice to reconstruct any unknown $n<^>{\times d}$ hypermatrix. In contrast to the earlier bound, our new exponent is bounded away from $1$ even as $d$ becomes very large.
A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory. We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$ ). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erd & odblac;s-R & eacute;nyi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n<^>{2/3})$ . A key component of this construction is our matrix-theoretic view of lifts. We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey.
Balister, the second author, Groenland, Johnston and Scott recently showed that there are asymptotically C4^n/n^3/4 many unordered sequences that occur as degree sequences of graphs. Combining limit theory for infinitely divisible distributions with a new bijective connection between a class of random walk trajectories and a subset counting formula from additive number theory, we describe C in terms of Walkup's number of rooted plane trees. The bijection is related to an instance of the Lévy-Khintchine formula. Our main result complements a result of Stanley, that ordered graphical sequences are related to quasi-forests.
We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving normal subgroups of quotients of the group, we show that the random walk converges to the uniform distribution on the group and give bounds for the convergence rate using spectral properties of the random walk steps. As an application, we use the moment method of Wood to prove a universality theorem for cokernels of random integer matrices allowing some dependence between entries.
We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+{\mathrm{o}}(n)$ . This can be seen as a directed graph analogue of a well-known theorem of Koml & oacute;s, S & aacute;rk & ouml;zy, and Szemer & eacute;di. Our result for trees follows from a more general result, allowing the embedding of arbitrary orientations of a much wider class of spanning 'tree-like' structures, such as collections of at most $O(n<^>{0.99})$ pairwise vertex-disjoint cycles and subdivisions of graphs $H$ with $|H|\lt \exp \bigl (\sqrt { extrm {O}(\log n)}\,\bigr )$ in which each edge is subdivided at least once.
Let Sigma be an alphabet and mu be a distribution on Sigma k for some k similar to 2. Let alpha > 0 be the minimum probability of a tuple in the support of mu (denoted supp(mu)). We treat the parameters Sigma, k, mu, alpha as fixed and constant. We say that the distribution mu has a linear embedding if there exist an Abelian group G (with the identity element 0G) and mappings sigma i: Sigma -> G, 1 <= i <= k, such that at least one of the mappings is non-constant and for every (a1, a2,..., ak). supp(mu), Sigma ki=1 sigma i(ai)= 0G. In [Bhangale-Khot-Minzer, STOC 2022], the authors asked the following analytical question. Let fi: Sigma n -> [- 1, 1] be bounded functions, such that at least one of the functions fi essentially has degree at least d, meaning that the Fourier mass of fi on terms of degree less than d is at most delta. If mu has no linear embedding (over any Abelian group), then is it necessarily the case that |E (x1,x2,...,xk)similar to mu circle times n [f1(x1)f2(x2) . . . fk(xk)]| = od,delta(1), where the right hand side. 0 as the degree d -> infinity and delta -> 0? In this paper, we answer this analytical question fully and in the affirmative for k= 3. We also show the following two applications of the result. 1. The first application is related to hardness of approximation. Using the reduction from [5], we show that for every 3-ary predicate P: Sigma 3 -> {0, 1} such that P has no linear embedding, an SDP (semi-definite programming) integrality gap instance of a P-Constraint Satisfaction Problem (CSP) instance with gap (1, s) can be translated into a dictatorship test with completeness 1 and soundness s + o(1), under certain additional conditions on the instance. 2. The second application is related to additive combinatorics. We show that if the distribution mu on Sigma 3 has no linear embedding, marginals of mu are uniform on Sigma, and (a, a, a) is an element of supp(mu) for every alpha is an element of Sigma, then every large enough subset of Sigma n contains a triple (x1, x2, x3) from mu circle times n (and in fact a significant density of such triples).
