By Christos A. Athanasiadis, Victor V. Batyrev, Dimitrios I. Dais, Martin Henk, and Francisco Santos

This quantity includes unique learn and survey articles stemming from the Euroconference "Algebraic and Geometric Combinatorics". The papers speak about quite a lot of difficulties that illustrate interactions of combinatorics with different branches of arithmetic, equivalent to commutative algebra, algebraic geometry, convex and discrete geometry, enumerative geometry, and topology of complexes and partly ordered units. one of the themes lined are combinatorics of polytopes, lattice polytopes, triangulations and subdivisions, Cohen-Macaulay mobilephone complexes, monomial beliefs, geometry of toric surfaces, groupoids in combinatorics, Kazhdan-Lusztig combinatorics, and graph colorations. This booklet is geared toward researchers and graduate scholars attracted to numerous features of recent combinatorial theories

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**Example text**

If α is an (n − 1 − i)-dimensional subspace of a polar space Γ of rank n (0 ≤ i ≤ n), then the set of all maximal subspaces through α deﬁnes a convex subspace of diameter i of the dual polar space S associated with Γ. Conversely, every convex subspace of S is obtained in this way. e. convex subspaces of diameter 2) are thick generalized quadrangles. Given a number of polar spaces, many others can be constructed. Let (Γi )i∈I be a family of polar spaces deﬁned on the sets (Pi )i∈I and with (Ai )i∈I as a collection of subspaces.

Suppose that any common neighbour of a and c lies at distance i − 1 from x. 21, there exists a path of length 3 in Γ2 (πQ (x)) ∩ Q connecting a and c. This path is completely contained in Γi (x). Deﬁnitions. If γ = (y0 , . . , yk ) denotes a path of S, then we deﬁne b(γ) := y0 and e(γ) := yk . For every point x of S, let Ωx denote the set of all paths (y0 , . . , yk ) in S for which (S(x, yi ) \ S(x, yi−1 )) ∩ S(x, y0 ) = ∅ for every i ∈ {1, . . , k} such that k d(x, yi ) > d(x, yi−1 ). For each such path γ, we deﬁne i(γ) := i=0 3d(x,y0 )−d(x,yi ) .

Let x be a given point of Q. For every point y ∈ Γ2 (x), we deﬁne A(y) := Γ1 (x) ∩ Γ1 (y) and A(y) := Γ1 (x) \ A(y). If y and y are two collinear points of Γ2 (x), then A(y) ∩ A(y ) consists of the unique point of yy collinear with x. 21, the diameter of Γ2 (x) is at most 3. So, if y and y are two points of Γ2 (x), we have one of the following possibilities: • y = y . Then |A(y) ∩ A(y )| = 0. • y ∼ y . Then |A(y) ∩ A(y )| = 1. • y and y have distance 2 in Γ2 (x). Let y denote a point of Γ2 (x) collinear with y and y .