4 problems

For integers $k\geq c\geq 2$, let $\operatorname{LO}_c$ and $\operatorname{LO}_k$ denote the relational structures whose ternary relation encodes linearly ordered $c$-colouring and…

For integers $k\ge 3$, let $f_1(e,q,k)$ denote the maximum chromatic number of a $1$-$(q,k)$-system with $e$ $q$-cliques. The asymptotic chromatic-number conjecture. Fix $k\ge 3$.…

Defective-colouring conjecture. Every $(r+1)$-uniform linear hypergraph with maximum degree at most $\Delta$ has a $d$-defective colouring with

Let $k,c\geq 2$. Define $A$ to be the relational structure on $E_{k+1}$ whose relation consists of the $k$-tuples with pairwise distinct entries, and define $B$ to be the relationa…