Bounds on sets with few distances

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of subsets; (2) we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances; (3) we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte. We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.Comment: 11 page

Cross-intersecting families and primitivity of symmetric systems

Let $X$ be a finite set and $\mathfrak p\subseteq 2^X$, the power set of $X$, satisfying three conditions: (a) $\mathfrak p$ is an ideal in $2^X$, that is, if $A\in \mathfrak p$ and $B\subset A$, then $B\in \mathfrak p$; (b) For $A\in 2^X$ with $|A|\geq 2$, $A\in \mathfrak p$ if $\{x,y\}\in \mathfrak p$ for any $x,y\in A$ with $x\neq y$; (c) $\{x\}\in \mathfrak p$ for every $x\in X$. The pair $(X,\mathfrak p)$ is called a symmetric system if there is a group $\Gamma$ transitively acting on $X$ and preserving the ideal $\mathfrak p$. A family $\{A_1,A_2,\ldots,A_m\}\subseteq 2^X$ is said to be a cross-$\mathfrak{p}$-family of $X$ if $\{a, b\}\in \mathfrak{p}$ for any $a\in A_i$ and $b\in A_j$ with $i\neq j$. We prove that if $(X,\mathfrak p)$ is a symmetric system and $\{A_1,A_2,\ldots,A_m\}\subseteq 2^X$ is a cross-$\mathfrak{p}$-family of $X$, then $$\sum_{i=1}^m|{A}_i|\leq\left\{ \begin{array}{cl} |X| & \hbox{if $m\leq \frac{|X|}{\alpha(X,\, \mathfrak p)}$,} \\ m\, \alpha(X,\, \mathfrak p) & \hbox{if $m\geq \frac{|X|}{\alpha{(X,\, \mathfrak p)}}$,} \end{array}\right.$$ where $\alpha(X,\, \mathfrak p)=\max\{|A|:A\in\mathfrak p\}$. This generalizes Hilton's theorem on cross-intersecting families of finite sets, and provides analogs for cross-$t$-intersecting families of finite sets, finite vector spaces and permutations, etc. Moreover, the primitivity of symmetric systems is introduced to characterize the optimal families.Comment: 15 page

Intersecting families of discrete structures are typically trivial

The study of intersecting structures is central to extremal combinatorics. A family of permutations $\mathcal{F} \subset S_n$ is \emph{$t$-intersecting} if any two permutations in $\mathcal{F}$ agree on some $t$ indices, and is \emph{trivial} if all permutations in $\mathcal{F}$ agree on the same $t$ indices. A $k$-uniform hypergraph is \emph{$t$-intersecting} if any two of its edges have $t$ vertices in common, and \emph{trivial} if all its edges share the same $t$ vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for $n$ sufficiently large with respect to $t$, the largest $t$-intersecting families in $S_n$ are the trivial ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest $t$-intersecting $k$-uniform hypergraphs are also trivial when $n$ is large. We determine the \emph{typical} structure of $t$-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollob\'as set-pairs inequality to bound the number of maximal intersecting families, which can then be combined with known stability theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira result. Update 2: corrected statement of the unpublished Hamm--Kahn result, and slightly modified notation in Theorem 1.6 Update 3: new title, updated citations, and some minor correction

Regular Intersecting Families

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the same (or approximately the same) number of members of $\mathcal{F}$. In particular, we show that we can guarantee $|\mathcal{F}| = o({n-1\choose k-1})$ if and only if $k=o(n)$.Comment: 15 pages, accepted versio

On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method

We study the function $M(n,k)$ which denotes the number of maximal $k$-uniform intersecting families $F\subseteq \binom{[n]}{k}$. Improving a bound of Balogh at al. on $M(n,k)$, we determine the order of magnitude of $\log M(n,k)$ by proving that for any fixed $k$, $M(n,k) =n^{\Theta(\binom{2k}{k})}$ holds. Our proof is based on Tuza's set pair approach. The main idea is to bound the size of the largest possible point set of a cross-intersecting system. We also introduce and investigate some related functions and parameters.Comment: 11 page