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Register a new userOn set systems without a simplex-cluster and the Junta method
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Noam Lifshitz
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A family ${A_{0},ldots,A_{d}}$ of $k$-element subsets of $[n]={1,2,ldots,n}$ is called a simplex-cluster if $A_{0}capcdotscap A_{d}=varnothing$, $|A_{0}cupcdotscup A_{d}|le2k$, and the intersection of any $d$ of the sets in ${A_{0},ldots,A_{d}}$ is nonempty. In 2006, Keevash and Mubayi conjectured that for any $d+1le klefrac{d}{d+1}n$, the largest family of $k$-element subsets of $[n]$ that does not contain a simplex-cluster is the family of all $k$-subsets that contain a given element. We prove the conjecture for all $kgezeta n$ for an arbitrarily small $zeta>0$, provided that $nge n_{0}(zeta,d)$. We call a family ${A_{0},ldots,A_{d}}$ of $k$-element subsets of $[n]$ a $(d,k,s)$-cluster if $A_{0}capcdotscap A_{d}=varnothing$ and $|A_{0}cupcdotscup A_{d}|le s$. We also show that for any $zeta nle klefrac{d}{d+1}n$ the largest family of $k$-element subsets of $[n]$ that does not contain a $(d,k,(frac{d+1}{d}+zeta)k)$-cluster is again the family of all $k$-subsets that contain a given element, provided that $nge n_{0}(zeta,d)$. Our proof is based on the junta method for extremal combinatorics initiated by Dinur and Friedgut and further developed by Ellis, Keller, and the author.
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The notion of cross intersecting set pair system of size $m$, $Big({A_i}_{i=1}^m, {B_i}_{i=1}^mBig)$ with $A_icap B_i=emptyset$ and $A_icap B_j eemptyset$, was introduced by Bollobas and it became an important tool of extremal combinatorics. His classical result states that $mle {a+bchoose a}$ if $|A_i|le a$ and $|B_i|le b$ for each $i$. Our central problem is to see how this bound changes with the additional condition $|A_icap B_j|=1$ for $i e j$. Such a system is called $1$-cross intersecting. We show that the maximum size of a $1$-cross intersecting set pair system is -- at least $5^{n/2}$ for $n$ even, $a=b=n$, -- equal to $bigl(lfloorfrac{n}{2}rfloor+1bigr)bigl(lceilfrac{n}{2}rceil+1bigr)$ if $a=2$ and $b=nge 4$, -- at most $|cup_{i=1}^m A_i|$, -- asymptotically $n^2$ if ${A_i}$ is a linear hypergraph ($|A_icap A_j|le 1$ for $i e j$), -- asymptotically ${1over 2}n^2$ if ${A_i}$ and ${B_i}$ are both linear hypergraphs.
Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of any t sets. The maximal size of such a set system is determined exactly if s+t<5, and asymptotically if s+t>4. Finally, we exactly determine the maximal size of a k-uniform set system that has the above described (s,t)-union-intersecting property, for large enough n.
In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra (one-sided) with basis all the faces of the simplices.
Alfeld introduced a subdivision AS(n) of an n-simplex, generalizing the Clough-Tocher split of a triangle. A formula for the dimension of the spline space C^r_k(AS(n)) was conjectured recently by Foucart-Sorokina. We prove that the graded module of C^r-splines on the cone over AS(n) is isomorphic to the module D^{r+1}(A_n) of multiderivations on the type A_n Coxeter arrangement. A theorem of Terao shows that the module of multiderivations of a Coxeter arrangement is free and gives an explicit basis. As a consequence the conjectured formula holds.
We are given a set $A$ of buyers, a set $B$ of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping $tau$ from $A$ to $B$, and $tau$ is strictly better than another house allocation $tau eq tau$ if for every buyer $i$, $tau(i)$ does not come before $tau(i)$ in the preference list of $i$. A house allocation is Pareto optimal if there is no strictly better house allocation. Let $s(tau)$ be the image of $tau$ (i.e., the set of houses sold in the house allocation $tau$). We are interested in the largest possible cardinality $f(m)$ of the family of sets $s(tau)$ for Pareto optimal mappings $tau$ taken over all sets of preference lists of $m$ buyers. We improve the earlier upper bound on $f(m)$ given by Asinowski, Keszegh and Miltzow by making a connection between this problem and some problems in extremal set theory.
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