Counting orientations of graphs with no strongly connected tournaments

Let Sk(n) be the maximum number of orientations of an n-vertex graph G in which no copy of Kk is strongly connected. For all integers n, k ≥ 4 where n ≥ 5 or k ≥ 5, we prove that Sk(n) = 2tk - 1(n), where tk-1(n) is the number of edges of the n-vertex (k - 1)-partite Turán graph Tk-1(n). Moreover, we prove that Tk-1(n) is the only graph having 2tk-1(n) orientations with no strongly connected copies of Kk

Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page