Tur\`an numbers of Multiple Paths and Equibipartite Trees

The Tur\'an number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erd\H{o}s-S\'os conjecture, and conditional on its truth we determine ex(n;H) when H is an equibipartite forest, for appropriately large n.Comment: 17 pages, 13 figures; Updated to incorporate referee's suggestions; minor structural change

On the definability of properties of finite graphs

AbstractThis paper considers the definability of graph-properties by restricted second-order and first-order sentences. For example, it is shown that the class of Hamiltonian graphs cannot be defined by monadic second-order sentences (i.e., if quantification over the subsets of vertices is allowed); any first-order sentence that defines Hamiltonian graphs on n vertices must contain at least 12n quantifiers. The proofs use Fraïssé-Ehrenfeucht games and ultraproducts