A local density condition for triangles

Discrepancy in Graphs and Hypergraphs

“…Surprisingly, the correct measure turns out to be the product disc + (G)disc − (G). We remark that a different type of negative discrepancy was considered by Erdős, Faudree, Rousseau and Schelp [5] with the idea of showing that graphs with small negative discrepancy contain complete subgraphs of fixed size. For further recent results in this direction see Krivelevich [9] and Keevash and Sudakov [8].…”

Sparse halves in triangle-free graphs

“…How large can β be as a function of α? Erdős, Faudree, Rousseau and Schelp [5] studied this problem and conjectured that there is a constant c t < 1 so that if c t α 1 then the largest possible β is t−2 t−1 (α − 1/2) (which is attained by the Turán graph T t−1 (n)). They proved this for triangle-free graphs (t = 3) and the general case was proved by the authors [7].…”

“…Moreover, for triangle-free graphs and general α it was conjectured in [5] that β is determined by a family of extremal triangle-free graphs. Besides the complete bipartite graph T 2 (n) already mentioned, another important graph is C 5 (n/5), which is obtained from a 5-cycle by replacing each vertex i by an independent set V i of size n/5 (assuming for simplicity that n is divisible by 5), and each edge ij by a complete bipartite graph joining V i and V j (this operation is called a 'blow-up').…”

Sparse halves in dense triangle-free graphs

“…, w 5 }. We define a collection {s i,j } i∈ [5],j∈ [4] of halves of (P * , ω). Fix i ∈ [5] and consider the vertex w i .…”