It is a folklore conjecture that the M\"obius function exhibits cancellation on shifted primes; is, $\sum_{p\le X}\mu(p+h) \ = o(\pi(X))$ as $X\to\infty$ for any fixed shift $h>0$. This appears in print at least since Hildebrand 1989. We prove average shifts $h\le H$, provided $\log H/\log\log X\to\infty$. also obtain results of prime $k$-tuples, and higher correlations with von Mangoldt diviso...

In 1977, Caccetta and Haggkvist [1] conjectured that if G is a directed graph with n vertices and if each vertex of G has outdegree at least k, then G contains a directed cycle of length at most In~k]. We shall refer to this conjecture as the C H conjecture. Trivially, this conjecture is true for k = 1, and it has been proved for k = 2 (Caccetta and Haggkvist [1]) and k = 3 (Hamildoune [3]). Ch...

This note is a brief introduction to theoretical and experimental results concerning Hilberg’s conjecture, a hypothesis about natural language. The aim of the text is to provide a short guide to the literature. 1 What is Hilberg’s conjecture? In the early days of information theory, Shannon (1951) published estimates of conditional entropy for printed English. A few decades later, Hilberg (1990...

There is some evidence for this conjecture. Mitchell [14] and Henn [10] have proved it for n ≤ 3. Voevodsky has announced a proof of the mod 2 QuillenLichtenbaum Conjecture for Z, and from [5] and [14] it follows that ι∗n is injective on the image of H∗(BGL(Λ);F2) −→ H ∗(BGn;F2). In particular, 1.1 is true in the stable range. The aim of this paper, though, is to give a disproof of Conjecture 1.1.

A beautiful conjecture of Erdős-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, ...

A family H of sets is said to be hereditary if all subsets of any set in H are in H; in other words, H is hereditary if it is a union of power sets. A family A is said to be intersecting if no two sets in A are disjoint. A star is a family whose sets contain at least one common element. An outstanding open conjecture due to Chvátal claims that among the largest intersecting sub-families of any ...

The Erdös-Hajnal conjecture states that for every graph H, there exists a constant δ(H) > 0 such that every graph G with no induced subgraph isomorphic to H has either a clique or a stable set of size at least |V (G)|. This paper is a survey of some of the known results on this conjecture.

Recent work of Shareshian and Wachs, Brosnan Chow, Guay-Paquet connects the well-known Stanley-Stembridge conjecture in combinatorics to dot action symmetric group $S_n$ on cohomology rings $H^*(Hess(S,h))$ regular semisimple Hessenberg varieties. In particular, order prove conjecture, it suffices construct (for any function $h$) a permutation basis whose elements have stabilizers isomorphic Yo...

The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools to prove the symmetric Jacobian Conjecture for the case F = X−H with H homogeneous and JH = 0. Other special results are also derived. We pose a combinatorial...