Abstract. A graph [Formula: see text] is common if its Ramsey multiplicity, i.e., the minimum number of monochromatic copies of [Formula: see text] contained in any 2-edge-coloring of [Formula: see text], is asymptotically the same as the number of monochromatic copies in the random 2-edge-coloring of [Formula: see text]. Erdős conjectured that every complete graph is common, which was disproved by Thomason in the 1980s. Until today, a classification of common graphs remains a wide open and challenging problem. Grzesik et al. [ Combin. Probab. Comput., 31 (2022), 907–923] conjectured that every [Formula: see text]-apex of any connected Sidorenko graph is common. We prove for [Formula: see text] that the [Formula: see text]-apex of any tree is common.

Abstract. Erdős, Lovász, and Spencer showed in the late 1970s that the dimension of the region of [Formula: see text]-vertex graph profiles, i.e., the region of feasible densities of [Formula: see text]-vertex graphs in large graphs, is equal to the number of nontrivial connected graphs with at most [Formula: see text] vertices. We determine the dimension of the region of [Formula: see text]-vertex tournament profiles. Our result, which explores an interesting connection to Lyndon words, yields that the dimension is much larger than just the number of strongly connected tournaments, which would be the answer expected as the analogy to the setting of graphs.