I will talk about the three objects and the two inclusions mentioned in the title from a tropical viewpoint. Possible generalizations will be speculated.

No Combsem talk this week -- NogaFest is on! http://www.math.tau.ac.il/~noga60/program.html

Speaker: Asaf Nachmias (TAU) Title: The connectivity of the uniform spanning forest on planar graphs Abstract: The free uniform spanning forest (FUSF) of an infinite connected graph G is obtained as the weak limit uniformly chosen spanning trees of finite subgraphs of G. It is easy to see that the FUSF is supported on spanning graphs of G with no cycles, but it need not be connected. Indeed, a classical result of Pemantle ('91) asserts that when G=Z^d, the FUSF is almost surely a connected tree if and only if d=1,2,3,4.

Speaker: Clara Shikhelman, TAU Title: Many T copies in H-free graphs. Abstract: For two graphs T and H and for an integer n, let ex(n,T,H) denote the maximum possible number of copies of T in an H-free graph on n vertices. The study of this function when T=K_2 (a single edge) is the main subject of extremal graph theory. We investigate the general function, focusing on the cases of triangles, complete graphs and trees. In this talk the main results will be presented as will sketches of proofs of some of the following: (i) ex(n,K_3,C_5) < (1+o(1)) (\sqrt 3)/2 n^{3/2}.

Title: Discrete Geometry in Minkowski Spaces Abstract: In recent decades, many papers appeared in which typical problems of Discrete Geometry are investigated, but referring to the more general setting of finite dimensional real Banach spaces (i.e., to Minkowski Geometry). In several cases such problems are investigated in the even more general context of spaces with so-called asymmetric norms (gauges). In many cases the extension of basic geometric notions, needed for posing these problems in non-Euclidean Banach spaces, is already interesting enough.

Speaker: Micha Sharir (Tel Aviv University) Title: Eliminating depth cycles for lines and triangles, with applications to bounding incidences Abstract: --------- The talk presents three related results. We first consider the problem of eliminating all depth cycles in a set of n lines in 3-space. For two lines l_1, l_2 in 3-space (in general position), we say that l_1 lies below l_2 if the unique vertical line that meets both lines meets l_1 at a point below the point where it meets l_2. This depth relationship typically has cycles, which can be eliminated if we cut the lines into

Speaker: Gil Kalai, HU Title: Algebraic-topological invariants of hypergraphs and extremal combinatorics Abstract: We will discuss some algebraic invariants of hypergraphs and some connection to algebraic topology. We will present some conjectural (rather speculative) relations with two central problems in extremal combinatorics: The Turan (4,3) conjecture and the Erdos-Rado sunflower conjecture.

Speaker: Lukas Kühne (University of Bonn) Title: Heavy hyperplanes in multiarrangements and their freeness Abstract: One of the central topics among the theory of hyperplane arrangements is their freeness. Terao's conjecture tries to link the freeness with the combinatorics of an arrangement. One of the few categories of arrangements which satisfy this conjecture consists of 3-dimensional arrangements with an unbalanced Ziegler restriction. This means that the arrangement contains a lot of hyperplanes intersecting in one single line

Speaker 1: Sria Louis Title: Asymptotically Almost Every 2r-regular Graph has an Internal Partition Abstract: An internal partition of a graph is a partitioning of the vertex set into two parts such that for every vertex, at least half of its neighbors are on its side. It is easy to notice that such a partition doesn't always exist (e.g. - cliques), though, both the existence and finding of such a partition - are open problems. Stiebitz (1996), responding to a problem of Thomassen (1983), made a breakthrough in this area, but the question and some interesting generalizations are still open.

Speaker: Zilin Jiang (Technion) Title: Relations between Tverberg points and central points Abstract: Given 3n lines in general position in the plane, it is always possible to partition them into n triples of lines so that all the triangles, formed by the triples, share a common point. This result is known back in 1988 by J.P. Roudneff. Strangely, in higher dimensions, it is only proved by Roman Karasev for n that is a prime power.

NOTE THE SPECIAL TIME: 11:00--12:30 Speaker: Eyal Ackerman, University of Haifa at Oranim Title: Coloring points with respect to squares Abstract: Is there an absolute constant $m$ such that any finite planar point set can be 2-colored such that every axis-parallel square that contains at least $m$ points contains points of both colors? I will discuss this problem and related ones.