Chromatically optimal rigid graphs
Journal of Combinatorial Theory Series B
Universal graphs without large cliques
Journal of Combinatorial Theory Series B
Regular Article: Universal Graphs with Forbidden Subgraphs and Algebraic Closure
Advances in Applied Mathematics
Duality theorems for finite structures (characterising gaps and good characterisations)
Journal of Combinatorial Theory Series B
The Homomorphism Structure of Classes of Graphs
Combinatorics, Probability and Computing
Journal of Graph Theory
Ramsey Classes and Homogeneous Structures
Combinatorics, Probability and Computing
Tree-depth, subgraph coloring and homomorphism bounds
European Journal of Combinatorics
European Journal of Combinatorics
Homomorphism preservation theorems
Journal of the ACM (JACM)
European Journal of Combinatorics
Generalised dualities and finite maximal antichains
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
On the tree-depth of random graphs
Discrete Applied Mathematics
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A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2ω the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended ("partnership theorem"). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets.This is related to universal graphs, rigid graphs (where we solve a problem posed in J. Combin. Theory B 46 (1989) 133) and to the density problem for countable graphs.