Note making a K 4-free graph bipartite

The well-known Max Cut problem asks for the largest bipartite subgraph of a graph G. This problem has been the subject of extensive research, both from the algorithmic perspective in computer science and the extremal perspective in combinatorics. Let n be the number of vertices and e be the number edges of G and let b(G) denote the size of the largest bipartite subgraph of G. The extremal part of Max Cut problem asks to estimate b(G) as a function of n and e. This question was first raised almost forty years ago by P. Erdős [8] and attracted a lot of attention since then (see, e.g., [3,2,4,1,16,11,10,5,7]). It is well known that every graph G with e edges can be made bipartite by deleting at most e/2 edges, i.e., b(G)≥e/2. To see this just consider a random partition of vertices of G into two parts V1,V2 and estimate the expected number of edges in the cut (V1,V2). A complete graphKn on n vertices shows