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

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 F of k-element subsets of [n] := {1, ..., n}, such that every element of [n] lies in the same (or approximately the same) number of members of.F. In particular, we show that we can guarantee vertical bar vertical bar = o(((n-1)(k-1))) if and only if k = o(n). (C) 2019 Published by Elsevier B.V

Conway groupoids, regular two-graphs and supersimple designs

A $2-(n,4,\lambda)$ design $(\Omega, \mathcal{B})$ is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym$(\Omega)$ called a "Conway groupoid". The construction generalizes Conway's construction of the groupoid $M_{13}$. It turns out that several infinite families of groupoids arise in this way, some associated with 3-transposition groups, which have two additional properties. Firstly the set of collinear point-triples forms a regular two-graph, and secondly the symmetric difference of two intersecting lines is again a line. In this paper, we show each of these properties corresponds to a group-theoretic property on the groupoid and we classify the Conway groupoids and the supersimple designs for which both of these two additional properties hold.Comment: 17 page

A Proof of the Cameron-Ku conjecture

A family of permutations A \subset S_n is said to be intersecting if any two permutations in A agree at some point, i.e. for any \sigma, \pi \in A, there is some i such that \sigma(i)=\pi(i). Deza and Frankl showed that for such a family, |A| <= (n-1)!. Cameron and Ku showed that if equality holds then A = {\sigma \in S_{n}: \sigma(i)=j} for some i and j. They conjectured a `stability' version of this result, namely that there exists a constant c < 1 such that if A \subset S_{n} is an intersecting family of size at least c(n-1)!, then there exist i and j such that every permutation in A maps i to j (we call such a family `centred'). They also made the stronger `Hilton-Milner' type conjecture that for n \geq 6, if A \subset S_{n} is a non-centred intersecting family, then A cannot be larger than the family C = {\sigma \in S_{n}: \sigma(1)=1, \sigma(i)=i \textrm{for some} i > 2} \cup {(12)}, which has size (1-1/e+o(1))(n-1)!. We prove the stability conjecture, and also the Hilton-Milner type conjecture for n sufficiently large. Our proof makes use of the classical representation theory of S_{n}. One of our key tools will be an extremal result on cross-intersecting families of permutations, namely that for n \geq 4, if A,B \subset S_{n} are cross-intersecting, then |A||B| \leq ((n-1)!)^{2}. This was a conjecture of Leader; it was recently proved for n sufficiently large by Friedgut, Pilpel and the author.Comment: Updated version with an expanded open problems sectio