Conjecture Of Kalai Research Articles

Reconstructing simplicial polytopes from their graphs and affine 2-stresses

A conjecture of Kalai from 1994 posits that for an arbitrary 2 ≤ k ≤ ⌊d/2⌋, the combinatorial type of a simplicial d-polytope P is uniquely determined by the (k − 1)-skeleton of P (given as an abstract simplicial complex) together with the space of affine k-stresses on P. We establish the first non-trivial case of this conjecture, namely, the case of k = 2. We also prove that for a general k, Kalai’s conjecture holds for the class of k-neighborly polytopes.

Forbidden vector‐valued intersections

We solve a generalised form of a conjecture of Kalai motivated by attempts to improve the bounds for Borsuk's problem. The conjecture can be roughly understood as asking for an analogue of the Frankl-Rodl forbidden intersection theorem in which set intersections are vector-valued. We discover that the vector world is richer in surprising ways: in particular, Kalai's conjecture is false, but we prove a corrected statement that is essentially best possible, and applies to a considerably more general setting. Our methods include the use of maximum entropy measures, VC-dimension, Dependent Random Choice and a new correlation inequality for product measures.

Hypergraphs Not Containing a Tight Tree with a Bounded Trunk

An $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so that every edge $e$ contains a vertex $v$ that does not belong to any preceding edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai personal communication published in Frankl and Füredi, J. Combin. Theory Ser. A, 45 (1987), pp. 226--262, generalizing the Erdös--Sós conjecture for trees, asserts that if $T$ is a tight $r$-tree with $t$ edges and $G$ is an $n$-vertex $r$-uniform hypergraph containing no copy of $T$, then $G$ has at most $\frac{t-1}{r}\binom{n}{r-1}$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree such that every edge of $T-T'$ has $r-1$ vertices in some edge of $T'$ and a vertex outside $T'$. For $r\ge 3$, the only nontrivial family of tight $r$-trees for which this conjecture has been proved is the family of $r$-trees with trunk size one in J. Combin. Theory Ser. A, 45 (1987), pp. 226--262. Our main result is an asymptotic version of Kalai's conjecture for all tight trees $T$ of bounded trunk size. This follows from our upper bound on the size of a $T$-free $r$-uniform hypergraph $G$ in terms of the size of its shadow. We also give a short proof of Kalai's conjecture for tight $r$-trees with at most four edges. In particular, for 3-uniform hypergraphs, our result on the tight path of length $4$ implies the intersection shadow theorem of Katona Acta Math. Acad. Sci. Hungar., 15 (1964), pp. 329--337.

Perfect Prismatoids are Lattice Delaunay Polytopes

A perfect prismatoid is a convex polytope P such that for every its facet F there exists a supporting hyperplane α k F such that any vertex of P belongs to either F or α. Perfect prismatoids concern with Kalai conjecture, that any centrally symmetric dpolytope P has at least 3 d non-empty faces and any polytope with exactly 3 d non-empty faces is a Hanner polytope. Any Hanner polytope is a perfect prismatoid (but not vice versa). A 0/1-polytope is a convex hull of some vertices of the d-dimensional unit cube. We prove that every perfect prismatoid is affinely equivalent to some 0/1-polytope of the same dimension. (And therefore every perfect prismatoid is a lattice polytope.) Let Λ be a lattice in R d and D be a polytope inscribed in a sphere B. Denote a boundary of B by ∂B and an interior of B by int B. The polytope D is a lattice Delaunay polytope if Λ∩int B = ∅ and D is a convex hull of Λ∩∂B. We prove that every perfect prismatoid is affinely equivalent to some lattice Delaunay polytope.

Counterexamples to Kalai's conjecture C

We provide two simple counterexamples to Kalai's Conjecture C and discuss our perspective on the implications for the prospect of large-scale fault-tolerant quantum computation.

Counterexamples to Kalai's conjecture C

We provide two simple counterexamples to Kalai's Conjecture C and discuss our perspective on the implications for the prospect of large-scale fault-tolerant quantum computation.

Socles of Buchsbaum modules, complexes and posets

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel's conjecture for the maximum value of the Euler characteristic of a 2 k-dimensional simplicial manifold on n vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number.

Empty Simplices of Polytopes and Graded Betti Numbers

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope and applying a result of Migliore and the author. This approach allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.

Face vectors of flag complexes

A conjecture of Kalai and Eckhoff that the face vector of an arbitrary flag complex is also the face vector of some particular balanced complex is verified.

Generic initial ideals and squeezed spheres

In 1988 Kalai constructed a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjecture about generic initial ideals of Stanley–Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai's conjecture, based on the fact that every squeezed ( d − 1 ) -sphere is the boundary of a certain d-ball, called a squeezed d-ball, generic initial ideals of Stanley–Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the other hand, we study the squeezing operation, which assigns to each Gorenstein* complex Γ having the weak Lefschetz property a squeezed sphere Sq ( Γ ) , and show that this operation increases graded Betti numbers.

Extension of Arrow's theorem to symmetric sets of tournaments

Arrow's impossibility theorem [K.J. Arrow, Social Choice and Individual Values, Wiley, New York, NY, 1951] shows that the set of acyclic tournaments is not closed to non-dictatorial Boolean aggregation. In this paper we extend the notion of aggregation to general tournaments and we show that for tournaments with four vertices or more any proper symmetric (closed to vertex permutations) subset cannot be closed to non-dictatorial monotone aggregation and to non-neutral aggregation. We also demonstrate a proper subset of tournaments that is closed to parity aggregation for an arbitrarily large number of vertices. This proves a conjecture of Kalai [Social choice without rationality, Reviewed NAJ Economics 3(4)] for the non-neutral and the non-dictatorial and monotone cases and gives a counter example for the general case.

Empty Simplices of Polytopes and Graded Betti Numbers

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope and applying a result of Migliore and the author. This approach allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.

Intersection homology of toric varieties and a conjecture of Kalai

We prove an inequality, conjectured by Kalai, relating the g-polynomials of a poly- tope P ,af ace F, and the quotient polytope P=F, in the case where P is rational. We introduce a new family of polynomials g(P;F), which measures the complexity of the part of P \far away from the face F ; Kalai's conjecture follows from the nonnegativity of these polynomials. This nonnegativity comes from showing that the restriction of the intersection cohomology sheaf on a toric variety to the closure of an orbit is a direct sum of intersection homology sheaves. Mathematics Subject Classication (1991). Primary 14F32; Secondary 14M25, 52B05.