On two problems in Ramsey-Tur\'an theory

Alon, Balogh, Keevash and Sudakov proved that the $(k-1)$-partite Tur\'an graph maximizes the number of distinct $r$-edge-colorings with no monochromatic $K_k$ for all fixed $k$ and $r=2,3$, among all $n$-vertex graphs. In this paper, we determine this function asymptotically for $r=2$ among $n$-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with sub-linear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an $n$-vertex $K_k$-free graph $G$ with $\alpha(G)=o(n)$. The extremal graphs have similar structure to the extremal graphs for the classical Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page

Boxicity of graphs on surfaces

The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface $\Sigma$ of genus $g$ is at most $5g+3$. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure

A note on forbidding clique immersions

Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without $K_t$-immersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without $\vec{K}_t$-immersions, where $\vec{K}_t$ denotes the bidirected complete digraph of order $t$