A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k+ 1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin [8, 9] asked whether k-existentially complete triangle-free graphs exist for every k. Here we present known and new constructions of 3existentially complete triangle-free ...

We investigate the automorphisms of some $$\kappa $$ -existentially closed groups. In particular, we prove that Aut(G) is union subgroups level preserving and $$|Aut(G)|=2^{\kappa }$$ whenever inaccessible G unique group cardinality . Indeed, latter result a byproduct an argument showing that, for any uncountable limit regular representation length with countable base, have $$|Aut(G)|=\beth _{\...

Temporal verification of universal (i.e., valid for all computation paths) properties of various kinds of programs, e.g., procedural, multi-threaded, or functional, can be reduced to finding solutions for equations in form of universally quantified Horn clauses extended with well-foundedness conditions. Dealing with existential properties (e.g., whether there exists a particular computation pat...

The quantified constraint satisfaction problem (QCSP) is a powerful framework for modelling computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified varia...

Pop proved that a smooth curve C over an ample field K with C(K) 6= ∅ has |K| many rational points. We strengthen this result by showing that there are |K| many rational points that do not lie in a given proper subfield, even after applying a rational map. As a consequence we gain insight into the structure of existentially definable subsets of ample fields. In particular, we prove that a perfe...