Abstract. We prove that for every set [Formula: see text] of vertices of a directed graph [Formula: see text], the maximum number of vertices in [Formula: see text] contained in a collection of vertex-disjoint cycles in [Formula: see text] is at least the minimum size of a set of vertices that hits all cycles containing a vertex of [Formula: see text]. As a consequence, the directed tree-width of a directed graph is linearly bounded in its cycle-width, which improves the previously known quadratic upper bound. We further show that the corresponding statement in bidirected graphs is true and that its edge-variant holds in both undirected and directed graphs, but fails in bidirected graphs. The vertex-version in undirected graphs remains an open problem.

Programs that combine I/O and countable probabilistic choice, modulo either bisimilarity or trace equivalence, can be seen as describing a probabilistic strategy. For well-founded programs, we might expect to axiomatize bisimilarity via a sum of equational theories and trace equivalence via a tensor of such theories. This is by analogy with similar results for nondeterminism, established previously. While bisimilarity is indeed axiomatized via a sum of theories, and the tensor is indeed at least sound for trace equivalence, completeness in general, remains an open problem. Nevertheless, we show completeness in the case that either the probabilistic choice or the I/O operations used are finitary. We also show completeness up to impersonation, i.e. that the tensor theory regards trace equivalent programs as solving the same system of equations. This entails completeness up to the cancellation law of the probabilistic choice operator.Furthermore, we show that a probabilistic trace strategy arises as the semantics of a well-founded program iff it is victorious. This means that, when the strategy is played against any partial counterstrategy, the probability of play continuing forever is zero.We link our results (and open problem) to particular monads that can be used to model computational effects.

Abstract It is shown that every model of NF admits a permutation model containing an internal automorphism.

Abstract We investigate Maker–Breaker games on graphs of size $\aleph _1$ in which Maker’s goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega ,\omega _1}$ -game under ZFC+MA+ $\neg $ CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega _1}$ -game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.

It is known that the cop number $c(G)$ of a connected graph $G$ can be bounded as a function of the genus of the graph $g(G)$. The best known bound, that $c(G) \leq \left\lfloor \frac{3 g(G)}{2}\right\rfloor + 3$, was given by Schröder, who conjectured that in fact $c(G) \leq g(G) + 3$. We give the first improvement to Schröder's bound, showing that $c(G) \leq \frac{4g(G)}{3} + \frac{10}{3}$.

Abstract We prove that for a finitely generated infinite nilpotent group G with structure (G, ·, …), the connected component G*0 of a sufficiently saturated extension G* of G exists and equals We construct an expansion of ℤ by a predicate (ℤ, +, P) such that the type-connected component is strictly smaller than ℤ*0. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem for finite partitions of groups.

Coproducts of monads on $\Set$ have arisen in both the study of computational effects and universal algebra. We describe coproducts of consistent monads on $\Set$ by an initial algebra formula, and prove also the converse: if the coproduct exists, so do the required initial algebras. That formula was, in the case of ideal monads, also used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent monads are injective; and that a coproduct of injective monad morphisms is injective. Two consistent monads have a coproduct iff either they have arbitrarily large common fixpoints, or one is an exception monad, possibly modified to preserve the empty set. Hence a consistent monad has a coproduct with every monad iff it is an exception monad, possibly modified to preserve the empty set. We also show other fixpoint results, including that a functor (not constant on nonempty sets) is finitary iff every sufficiently large cardinal is a fixpoint.

It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G. The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).