We describe several exotic fusion systems related to the sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow 3-subgroups of the Conway group $\textrm{Co}_1$ and the Thompson group $\textrm{F}_3$, and a Sylow 5-subgroup of the Monster M, as well as a particular maximal subgroup of the latter two p-groups. This work is supported by computations in MAGMA.

In this paper, we establish homological Berglund–Hübsch mirror symmetry for curve singularities where the A–model incorporates equivariance, otherwise known as homological Berglund–Hübsch–Henningson mirror symmetry, including for certain deformations of categories. More precisely, we prove a conjecture of Futaki and Ueda which posits that the equivariance in the A–model can be incorporated by pulling back the superpotential to the total space of the corresponding crepant resolution. Along the way, we show that the B–model category of matrix factorisations has a tilting object whose length is the dimension of the state space of the Fan–Jarvis–Ruan–Witten (FJRW) A–model, a result which might be of independent interest for its implications in the Landau–Ginzburg analogue of Dubrovin’s conjecture.

Given a fixed k-uniform hypergraph F, the F-removal lemma states that every hypergraph with few copies of F can be made F-free by the removal of few edges. Unfortunately, for general F, the constants involved are given by incredibly fast-growing Ackermann-type functions. It is thus natural to ask for which F one can prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when F is k-partite. Alon proved that when $k=2$ (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and Rödl conjectured in 2002 that Alon’s result can be extended to all $k\gt2$, namely, that the only $k$-graphs $F$ for which the hypergraph removal lemma has polynomial bounds are the trivial cases when F is k-partite. In this paper we prove this conjecture.

Let us say that a graph $G$ is Ramsey for a tuple $(H_1,\ldots,H_r)$ of graphs if every r-colouring of the edges of G contains a monochromatic copy of $H_i$ in colour i, for some $i \in [\![{r}]\!]$. A famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and Ruciński, predicts the threshold at which the binomial random graph $G_{n,p}$ becomes Ramsey for $(H_1,\ldots,H_r)$ asymptotically almost surely.

In this paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of graphs. Moreover, we reduce its validity to the truth of a certain deterministic statement, which is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs $H_1,\ldots,H_r$ by finite families $\mathcal{H}_1,\ldots,\mathcal{H}_r$. Additionally, we pose a natural (deterministic) graph-partitioning conjecture, which we believe to be of independent interest, and whose resolution would imply the Kohayakawa–Kreuter conjecture.