The following very natural problem was raised by Chung and Erd\H{o}s in the early 80's has since been repeated a number of times. What is minimum Tur\'an $\text{ex}(n,\mathcal{H})$ among all $r$-graphs $\mathcal{H}$ with fixed edges? Their actual focus on an equivalent perhaps even more question which asks what largest size $r$-graph that can not be avoided any $n$ vertices $e$ In original pape...

Let Fk denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollobás conjectured that for every 2 > 0 and positive integer k there is an n(k, 2) such that every 2-edge-coloring of the complete graph of order n ≥ n(k, 2) which has at least 2n2 ) edges in each color contains a member of Fk. This ...

An n × n array is avoidable if for each set of n symbols there is a Latin square on these symbols which differs from the array in every cell. We characterise all unavoidable square arrays with at most 2 symbols, and all unavoidable arrays of order at most 4. We also identify a number of general families of unavoidable arrays, which we conjecture to be a complete account of unavoidable arrays. N...

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