Yet another generalization of the Kruskal-Katona theorem

AbstractFor an n-tuple t = (t1,t2,…,tn) of integers satisfying 1⩽t1⩽t2···⩽tn, T(t)=T denotes the ranked partially ordered set consisting of n-tuples a = (a1,a2,…,an) of integers satisfying tn−ti⩽ai⩽tn, i = 1,2,…,n, partially ordered by defining a to precede c if ai = ci or ci = tn for i = 1,2,…,n. The rank r(a) of a is |{i|ai = tn}|. For 0⩽l⩽n, the set consisting of all elements of rank l is called the lth rank and is denoted Tl. Let b, l and m denote positive integers satisfying b⩽l⩽n and m⩽|Tl|. For a subset A of Tl, Δb A denotes the elements of Tl-b which precede at least one element of A. An algorithm is given for calculating min |Δb A|, where the minimum is taken over all m-element subsets A of Tl. If t1 = t2 = ··· = tn = 1, it reduces to the Kruskal-Katona algorithm

Kruskal--Katona-Type Problems via Entropy Method

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call this type of problems Kruskal--Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a $3$-colored graph with $R$ red, $G$ green, $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$, which is sharp. Second, we give a generalization of the Kruskal--Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.Comment: 18 page

Chip firing and all-terminal network reliability bounds

AbstractThe (all-terminal) reliability of a graph G is the probability that all vertices are in the same connected component, given that vertices are always operational but edges fail independently each with probability p. Computing reliability is #P-complete, and hence is expected to be intractable. Consequently techniques for efficiently (and effectively) bounding reliability have been the major thrust of research in the area. We utilize a deep connection between reliability and chip firings on graphs to improve previous bounds for reliability

The lex-plus-power inequality for local cohomology modules

We prove an inequality between Hilbert functions of local cohomology modules supported in the homogeneous maximal ideal of standard graded algebras over a field, within the framework of embeddings of posets of Hilbert functions. As a main application, we prove an analogue for local cohomology of Evans' Lex-Plus-Power Conjecture for Betti numbers. This results implies some cases of the classical Lex-Plus-Power Conjecture, namely an inequality between extremal Betti numbers. In particular, for the classes of ideals for which the Eisenbud-Green-Harris Conjecture is currently known, the projective dimension and the Castelnuovo-Mumford regularity of a graded ideal do not decrease by passing to the corresponding Lex-Plus-Power ideal.Comment: 15 pages, 1 figur