We generalize HOP by allowing tables of size two, rather than a minimum size of four as previously defined in HOP. Thus, in the generalized HOP, we aim to seat the $2n$ participants at $s$ tables of size $2$ and $t$ round tables of sizes $2m_1, 2m_2, \dots, 2m_t$, with the requirement that $2n = 2s + 2m_1 + 2m_2 + \dots + 2m_t$ and $m_i\geq 2$. Our current goal is to prove that the necessary condition for the HOP with \(s\) tables of size 2 and one large table of size \(2m\) to have a solution is sufficient. In this talk, we will present a general approach to this problem and discuss the progress we have made so far.

A Kirkman triple system consists of a resolvable Steiner triple system together with a partition of its blocks into parallel classes. We show that for every integer $\delta \geq 3$, there exist infinitely many $\delta$-chromatic Kirkman triple systems. Moreover, in the case $\delta=3$, we give a complete existence result for $3$-chromatic Kirkman triple systems.

This is joint work with Nicholas Cavenagh, Peter Danziger and David Pike.

In the special case $A=B$, the additive triple is called a \emph{Schur triple}. It is known that the Cauchy-Davenport Theorem gives bounds on the number $r(A,A,A)$ of Schur triples, and that the lower and upper bound can each be attained by a set $A$ that is an interval of $s$ consecutive elements of $\mathbb{Z}_p$. However, it is known that there are values of $p,s$ for which not every value from the lower bound to the upper bound is attainable.

In the case where $A,B$ can be distinct, we use Pollard's generalization of the Cauchy-Davenport Theorem to derive bounds on the possible values of the number $r(A,B,B)$ of additive triples. In contrast to the case $A=B$, we show that every value from the lower bound to the upper bound is attainable; and it is sufficient to take $B$ to be an interval of $t$ consecutive elements of~$\mathbb{Z}_p$.

This is joint work with Sophie Huczynska and Laura Johnson.

(i) There exists an Orthogonal Array OA$(n+1, n)$ and an $n \times (n+1)$ partial Hadamard matrix.

(ii) There exists a balancedly multi-splittable $n^2 \times n(n+1)$ partial Hadamard matrix.

Additionally, the concept of balancedly multi-splittable Balanced Incomplete Block Designs will be introduced and discussed.

A joint work with Sho Suda and Yash Khobragade.

We extend the definition of $2$-CFF to include a graph($G$), called $\overline{G}$-CFF, where the edges of $G$ specify the pair of subsets whose union must not cover any other subset. We denote by $t(G)$ as the minimum $t$ for which there exists a $\overline{G}$-CFF. Thus, $t(K_n) = t(2, n)$. We will discuss some classical results on CFFs, along with constructions of $\overline{G}$-CFFs. We prove that for a graph $G$ with $n$ vertices, $t(1, n) \leq t(G) \leq t(2, n)$ and for an infinite family of star graphs with $n$ vertices, $t(S_n) = t(1, n)$. We also provide constructions for $\overline{P_n}$-CFF and $\overline{C_n}$-CFF using a mixed-radix Gray code. This yields an upper bound for $t(P_n)$ and $t(C_n)$ that is smaller than the lower bound of $t(2, n)$ mentioned above.

Joint work with Lucia Moura.

Joint work with Benjamin Stanley.

Raaphorst, Moura, and Stevens (2014) give a construction for a CA$(2q^3-1; 3, q^2 + q + 1, q)$, for any prime power $q$. This is obtained by two projective planes $PG(2,q)$ such that any three collinear points in one is mapped to three non-collinear points in the other.

A $k$-cap of $PG(m-1, q)$ is a set of $k$ points no three of which are collinear. In $PG(3,q)$, an ovoid is a $k$-cap with maximum size of $k$. In a paper by Tzanakis, Moura, Panario, and Stevens (2016), a CA$(511; 4, 17, 4)$ is constructed, which was formed by two ovoids in $PG(3, 4)$ such that any four coplanar points in one is mapped to four non-coplanar points in the other.

In this talk, we give a construction for strength-$4$ covering arrays using three $k$-caps in $PG(3,q)$, which has been verified for all odd prime powers $q$ such that $3 \leq q \leq 101$. We conjecture that our construction is valid for any odd prime power $q$. This is joint work with Lucia Moura.