No Dense Subgraphs Appear in the Triangle-free Graph Process

The Final Size of theC4-Free Process

“…For random graph G(n, i) to have the property that the addition of any new edge creates a copy of H, provided H is strictly 2-balanced (see [12]). It is also known (see [7] and [14]) that sufficiently dense subgraphs are unlikely to appear in the final graph G(M(H)).…”

Dense subgraphs in the H-free process

“…Theorem 2). For the special case H = C 3 , Gerke and Makai [8] have previously obtained a similar result for fixed graphs F . So, Corollary 4 not only generalizes the main result of [8], but moreover demonstrates that whp dense graphs F never appear in the H-free process, also if their number of vertices grow moderately in n. In fact, we believe that fixed graphs with maximum density strictly larger than d 2 (H) do not appear in the H-free process.…”

“…Intuitively, we show that whp for every possible placement of ⌈c|A|⌉ edges inside some set A ⊆ [n] satisfying |A| ≤ n d , already after the first m steps there exists a 'witness' which certifies that not all of these edges can appear in the H-free process. The same basic idea was used in [8], but the main part of their argument is tailored towards the (simpler) C 3 -free case (in fact, a similar idea has also previously been used for bounding the independence number of the H-free process in [1,2]). By contrast, our argument is for the more general H-free process, where H is strictly 2-balanced, and one important ingredient are the estimates obtained by Bohman and Keevash [2].…”

“…Since precise structural properties of the final graph are not known so far, we do not ask whether an analogue of Theorem 1 also holds in the later evolution, but restrict our attention to the basic question whether fixed H-free graphs F satisfying, say, m(F ) ≫ d 2 (H) appear or not. For the special case H = C 3 this has recently been addressed by Gerke and Makai [8]: they proved that for some c > 0, whp fixed graphs F with e F /v F ≥ c do not appear in the C 3 -free process. We would like to remark that such a behaviour is not necessarily true for all constrained graph processes.…”

The diamond‐free process

“…(Similar results for the triangle-free process were obtained by Wolfovitz [14]). Extending results of Gerke and Makai [5] for the triangle-free process, Warnke [10] showed that there are constants c 1 , c 2 depending only on H so that, w.h.p., the graph G M (H) contains no subgraphs on at most n c 1 vertices with density at least c 2 . As a result of our analysis, one can modify Warnke's approach to show the existence of constants c 1 and c 2 so that the same conclusion holds for G M (K − 4 ) ; we leave these details to the interested reader.…”