Probabilistic existence of regular combinatorial structures

We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen structure has the required properties with positive yet tiny probability. Our method allows also to give rather precise estimates on the number of objects of a given size and this is applied to count the number of orthogonal arrays, t-designs and regular hypergraphs. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.Comment: An extended abstract of this work [arXiv:1111.0492] appeared in STOC 2012. This version expands the literature discussio

Probabilistic Existence of Large Sets of Designs

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been recentlyintroduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t$-$(n,k,\lambda)$ combinatorial design with tiny, yet positive, probability. The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, not only does a $t$-$(n,k,\lambda)$ design exist but, in fact, with positive probability there exists a large set of such designs -- that is, a partition of the set of $k$-subsets of $[n]$ into $t$-designs $t$-$(n,k,\lambda)$ designs. Specifically, using the probabilistic approach derived herein, we prove that for all sufficiently large $n$, large sets of $t$-$(n,k,\lambda)$ designs exist whenever $k > 12t$ and the necessary divisibility conditions are satisfied. This resolves the existence conjecture for large sets of designs for all $k > 12t$.Comment: 20 page

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

Nontrivial t-Designs over Finite Fields Exist for All t

A $t$-$(n,k,\lambda)$ design over \F_q is a collection of $k$-dimensional subspaces of \F_q^n, called blocks, such that each $t$-dimensional subspace of \F_q^n is contained in exactly $\lambda$ blocks. Such $t$-designs over \F_q are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,\lambda)$ designs over \F_q are currently known to exist only for $t \leq 3$. Herein, we prove that simple (meaning, without repeated blocks) nontrivial $t$-$(n,k,\lambda)$ designs over \F_q exist for all $t$ and $q$, provided that $k > 12t$ and $n$ is sufficiently large. This may be regarded as a $q$-analog of the celebrated Teirlinck theorem for combinatorial designs

Derandomized Construction of Combinatorial Batch Codes

Combinatorial Batch Codes (CBCs), replication-based variant of Batch Codes introduced by Ishai et al. in STOC 2004, abstracts the following data distribution problem: $n$ data items are to be replicated among $m$ servers in such a way that any $k$ of the $n$ data items can be retrieved by reading at most one item from each server with the total amount of storage over $m$ servers restricted to $N$. Given parameters $m, c,$ and $k$, where $c$ and $k$ are constants, one of the challenging problems is to construct $c$-uniform CBCs (CBCs where each data item is replicated among exactly $c$ servers) which maximizes the value of $n$. In this work, we present explicit construction of $c$-uniform CBCs with $\Omega(m^{c-1+{1 \over k}})$ data items. The construction has the property that the servers are almost regular, i.e., number of data items stored in each server is in the range $[{nc \over m}-\sqrt{{n\over 2}\ln (4m)}, {nc \over m}+\sqrt{{n \over 2}\ln (4m)}]$. The construction is obtained through better analysis and derandomization of the randomized construction presented by Ishai et al. Analysis reveals almost regularity of the servers, an aspect that so far has not been addressed in the literature. The derandomization leads to explicit construction for a wide range of values of $c$ (for given $m$ and $k$) where no other explicit construction with similar parameters, i.e., with $n = \Omega(m^{c-1+{1 \over k}})$, is known. Finally, we discuss possibility of parallel derandomization of the construction