Large Multipartite Subgraphs in H-free Graphs

10 Problems for Partitions of Triangle-free Graphs

“…The conjecture was verified for r = 5 by Hu, Lidický, Martins, Norin, and Volec [13] using flag algebras. The conjecture for even r seems to be more difficult than for odd r.…”

“…This idea comes from Naves [19] who proposed it at the Graduate Research Workshop in Combinatorics. It was also used by Hu, Lidický, Martins, Norin, and Volec [13] and by the authors in [2].…”

Making $K_{r+1}$-Free Graphs $r$-partite

“…Füredi [8] gave a nice short proof of the statement that a K r+1 -free graph G on n vertices with at least ex(n, K r+1 ) − t edges satisfies D r (G) ≤ t; thus providing a quantitative version of the Erdős-Simonovits theorem. In [10] Füredi's result was strengthened for some values of r. For small t, we will determine asymptotically how many edges are needed. For very small t, it is already known [3] that G has to be r-partite.…”

“…A related question was studied by Sudakov; he determined the maximum number of edges in a K 4 -free graph which need to be removed in order to make it bipartite [13]. This problem for K 6 -free graphs was solved by Hu, Lidický, Martins, Norin and Volec [10]. We will study the question of how many edges in a K r+1 -free graph need at most to be removed to make it r-partite.…”

Max Cuts in Triangle-free Graphs

“…Sudakov studied a related question; he [12] determined the maximum number D 2 (G) for K 4 -free graph G. Recently, Hu, Lidický, Martins, Norin and Volec [7] announced a proof for determining the maximum number D 2 (G) for n-vertex K 6free graphs G. They use the method of flag algebras, developed by Razborov [11], to describe local cuts which leads to the solution. We use a similar idea of encoding local cuts.…”