We investigate the following conjecture of Hehui Wu: for every tournament S, the class of S-free tournaments has bounded domination number. We show that the conjecture is false in general, but true when S is 2-colourable (that is, its vertex set can be partitioned into two transitive sets); the latter follows by a direct application of VC-dimension. Our goal is to go beyond this; we give a non-...

Let X be a variety of logarithmic general type, defined over a number field K. Let S be a finite set of places in K and let OK,S be the ring of S-integers. Suppose that X is a model of X over Spec OK,S . As a natural generalizasion of theorems of Siegel and Faltings, It was conjectured by S. Lang and P. Vojta ([Vojta], conjecture 4.4) that the set of S-integral points X (OK,S) is not Zariski de...

In this paper we conjecture that the piecewise linear map f(x) = px1I[0,1/p)(x)+ (sx − s/p)1I[1/p,1](x), p > 1, 0 < s < 1 which has an expanding, onto branch and a contracting branch is eventually piecewise expanding. We give a partial proof of the conjecture, in particular for values of p and s such that d− ln(p(1−s)+s) ln s e 6 = d− ln p ln s e.

0.1. The Moore conjecture and the Barratt conjecture. The fundamental problem in homotopy theory is how to determine homotopy groups. People found that it is very difficult to compute homotopy groups of finite complexes. So then we intend to know what we can say about homotopy groups, that is, how to study properties of homotopy groups instead of explicit calculations. Two famous conjectures on...

Let I be a monomial ideal in the polynomial ring S = K[x1, . . . , xn]. We study the Stanley depth of the integral closure I of I. We prove that for every integer k ≥ 1, the inequalities sdepth(S/Ik) ≤ sdepth(S/I) and sdepth(Ik) ≤ sdepth(I) hold. We also prove that for every monomial ideal I ⊂ S there exist integers k1, k2 ≥ 1, such that for every s ≥ 1, the inequalities sdepth(S/I1) ≤ sdepth(S...

A CIS-graph is defined as a graph whose every maximal clique and stable set intersect. These graphs have many interesting properties, yet, it seems difficult to obtain an efficient characterization and/or polynomial-time recognition algorithm for CIS-graphs. An almost CIS-graph is defined as a graph that has a unique pair (C, S) of disjoint maximal clique C and stable sets S. We conjecture that...

Given an abelian, CM extension K of any totally real number field k, we restate and generalise two conjectures ‘of Stark type’ made in [So5]. The Integrality Conjecture concerns the image of a p-adic map sK/k,S determined by the minus-part of the S-truncated equivariant L-function for K/k at s = 1. It is connected to the Equivariant Tamagawa Number Conjecture of Burns and Flach. The Congruence ...

Artin's celebrated conjecture on primitive roots (Artin [l, p. viii], Hasse [2], Hooley [3]) suggests the following Conjecture. Let S' be a set of rational primes. For each q£-S, let Lq be an algebraic number field of degree n(q). For every square-f ree integer k, divisible only by primes of S, define Lk to be the composite of all Lq, q\ k, and denote n{k) =deg(Ljb/0). Assume that 2 * l/n(k) co...

Let S be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of S having minimal depth. In particular, Stanley’s conjecture holds for these ideals. Also we show that if I is a monomial ideal with Ass(S/I) = {P1, P2, . . . , Ps} and Pi 6⊂ ∑s 1=j 6=i Pj for all i ∈ [s], then Stanley’s conjecture holds for S/

Ohba’s conjecture states that if a graph G has chromatic number χ(G) = k and has at most 2k+1 vertices, then G has choice number Ch(G) equal to χ(G). We prove that Ohba’s conjecture is true for each graph that has independence number 3. We also prove that Ohba’s conjecture is true for each graph G that has an independent set S of size 4 such that G\S has independence number 3.