A generalized Macaulay theorem and generalized face rings

We prove that the f-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0?k^∂(fk)?fk−1 for all k?0. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the “diamond property,” discussed by Wegner [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828].For geometric meet semi-lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which also includes multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.     

Intersecting antichains and shadows in linear lattices

We establish a homomorphism of finite linear lattices onto the Boolean lattices via a group acting on linear lattices. By using this homomorphism we prove the intersecting antichains in finite linear lattices satisfy an LYM-type inequality, as conjectured by Erd?s, Faigle and Kern, and we state a Kruskal-Katona type theorem for the linear lattices.     

A Splitting Inequality

We prove that if A is a finite set, and if is an downwards-closed family of subsets of A, and if fx is the proportion of x-element subsets of A in , then fa · fb fa + b – r, if r < . We connect this result with the Weak Threshold Theorem.     

Shadows and intersections in vector spaces

We prove a vector space analog of a version of the Kruskal-Katona theorem due to Lovász. We apply this result to extend Frankl's theorem on r-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erd?s-Ko-Rado theorem for vector spaces.     

Shadows of ordered graphs

Isoperimetric inequalities have been studied since antiquity, and in recent decades they have been studied extensively on discrete objects, such as the hypercube. An important special case of this problem involves bounding the size of the shadow of a set system, and the basic question was solved by Kruskal (in 1963) and Katona (in 1968). In this paper we introduce the concept of the shadow ∂G of a collection G of ordered graphs, and prove the following, simple-sounding statement: if n∈N is sufficiently large, |V(G)|=n for each G∈G, and |G|<n, then |∂G|?|G|. As a consequence, we substantially strengthen a result of Balogh, Bollobás and Morris on hereditary properties of ordered graphs: we show that if P is such a property, and |Pk|<k for some sufficiently large k∈N, then |Pn| is decreasing for k?n<∞.