We provide an independent resolution of Lam's problem by reducing the problem to a satisfiability (SAT) problem in Boolean logic that we solve with a combination of SAT solvers and computer algebra systems. Our resolution provides a collection of certificates that a third party can use to verify the nonexistence of a projective plane of order ten.

This is joint work with Peter Danziger and Tommaso Traetta.

We give a complete solution in the case when there are exactly three cycles of lengths $\alpha, \beta, \gamma \neq 2$. Somewhat surprisingly, the general necessary conditions turn out not to be sufficient in this case. In particular, taking $\gamma \geq \beta \geq \alpha > 2$, we show that when $\gamma=n$, $\alpha+\beta > n+2$ and $\alpha+\beta \equiv n$ (mod 4), $K_n^*$ is not decomposable.

Joint work with A.C. Burgess and M.T. Javed

This talk will discuss the currently known existence results of pairwise balanced designs PBD$(v,K)$ of dimension three, for the sets of block sizes $K=\{3,4\}$, and $K=\{3,5\}$. Also to be discussed is dimension three triple systems of arbitrary index, whose existence is a consequence of the existence of the aforementioned pairwise balanced designs.\

This is based on work with Peter Dukes, extending previous work by Dukes and Joanna Niezen.

In this talk, we shall consider the generalization of this conjecture to cyclic $(v,k,\lambda)$-designs with $1 \leq \lambda \leq k-1$. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all $v$, but we conjecture that it holds whenever $v$ is sufficiently large compared to $k$. We confirm that the generalization of the conjecture holds when $v$ is prime and $\lambda=1$ and also when $\lambda \leq (k-1)/2$ and $v$ is sufficiently large compared to $k$. As a corollary, we show that for any $k \geq 3$, with the possible exception of finitely many composite orders $v$, every cyclic $(v,k,1)$-design without short orbits is generated by a $(v,k,1)$-disjoint difference family.

This is joint work with Daniel Horsley and Xiaomiao Wang.

The talk is based on joint work with Daniela Kühn, Allan Lo and Deryk Osthus.

A famous conjecture, variously attributed to Ryser, Brualdi, and Stein, asserts that in every latin square of order $n$ it is possible to find a collection of $n-1$ cells which intersects each row and column, and contains each symbol, at most once---such a collection of cells is known as a near transversal. In this talk we will outline the proof that group-based latin squares satisfy the Ryser-Brualdi-Stein conjecture. We will then explain why the resolution of this special case is far from the end of the discussion of near transversals in group-based latin squares.

An $(n,k,1)$-design is a collection of $k$-subsets (blocks) of a set of $n$ points such that each pair of points occur together in exactly one block. If this restriction is relaxed to require only that each pair of points occur together in at most one block we instead have a partial $(n,k,1)$-design. I will outline a proof that any partial $(n,k,1)$-design with at most $\frac{n-1}{k-1}-k+1$ blocks is completable to a $(n,k,1)$-design provided that $n$ is sufficiently large and obeys the obvious necessary conditions for an $(n,k,1)$-design to exist. This result is tight for all $k$. I will also mention some related results concerning edge decompositions of almost complete graphs into copies of $K_k$. All of this work is joint with Ajani De Vas Gunasekara.

This is joint work with Thomas Pender and Sho Suda. }

In this talk, I will describe applications of our existence results for incomplete pairwise balanced designs, IPBD$((v;w),K)$. IPBDs provide a surprisingly powerful tool for constructing designs with substructures. We describe two basic techniques for applications: using an IPBD as a `template' in constructions and using IPBDs instead of PBDs in PBD-closure. Several examples will be given to illustrate the power of these techniques. These examples include using IPBDs to construct incomplete mutually orthogonal latin squares and resolvable designs with sub-designs. All of this work is joint work with Peter Dukes.

The EKR theorem can be extended to various combinatorial objects. In this presentation, I will talk about an extension of the EKR theorem to $2$-intersecting families of perfect matchings. In particular, I prove that any $2$-intersecting family of perfect matchings of $K_{2k}$ has size at most $(2k-5)\times (2k-7) \times \ldots 3\times 1$, for any $k\geq 3$.

This is joint work with A.C. Burgess, P. Danziger, J.H. Dinitz and D.M. Donovan.

In this talk, we shall give a thorough introduction to layering, and then describe new bipartite 2-factorizations of complete multigraphs obtained by layering. In particular, for complete multigraphs and bipartite 2-factors with no 2-cycles, we obtain a complete solution to the Oberwolfach Problem and an almost complete solution to the Hamilton-Waterloo Problem.

This is joint work with Amin Bahmanian.