We study the relationship between the sizes of two sets B, S ⊂ R 2 when B contains either the whole boundary, or the four vertices, of a square with axes-parallel sides and center in every point of S, where size refers to one of cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprinsingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point [0, 1] 2 , but prove that such a B must have packing and lower box dimension at least 7 4 , and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.

For a given finite poset P , La(n, P ) denotes the largest size of a family F of subsets of [n] not containing P as a weak subposet. We exactly determine La(n, P ) for infinitely many P posets. These posets are built from seven base posets using two operations. For arbitrary posets, an upper bound is given for La(n, P ) depending on |P | and the size of the longest chain in P . To prove these theorems we introduce a new method, counting the intersections of F with double chains, rather than chains.

The vertex set of the Kneser graph K(n, k) is V = [n] k and two vertices are adjacent if the corresponding sets are disjoint. For any graph F , the largest size of a vertex set U ⊆ V such that K(n, k)[U ] is F -free, was recently determined by Alishahi and Taherkhani, whenever n is large enough compared to k and F . In this paper, we determine the second largest size of a vertex set W ⊆ V such that K(n, k)[W ] is F -free, in the case when F is an even cycle or a complete multi-partite graph. In the latter case, we actually give a more general theorem depending on the chromatic number of F . Mathematics Subject Classification: 05C35, 05D05

We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges.An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending on the edge density. An easy lower bound is alsoproved. This answer resembles the classic theorem of Ahlswede and Katona about the maximal number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun about k-edge stars.

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of any t sets. The maximal size of such a set system is determined exactly if s + t ≤ 4, and asymptotically if s + t ≥ 5. Finally, we exactly determine the maximal size of a k-uniform set system that has the above described (s, t)-union-intersecting property, for large enough n.