Books versus triangles at the extremal density

A celebrated result of Mantel shows that every graph on $n$ vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must in fact be at least $\lfloor n/2 \rfloor$ triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erd\H{o}s, says that any such graph must have an edge which is contained in at least $n/6$ triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any $1/6 \leq \beta 0$ such that any graph on $n$ vertices with at least $\lfloor n^2/4\rfloor + 1$ edges and book number at most $\beta n$ contains at least $(\gamma -o(1))n^3$ triangles. He also asked for a more precise estimate for $\gamma$ in terms of $\beta$. We make a conjecture about this dependency and prove this conjecture for $\beta = 1/6$ and for $0.2495 \leq \beta < 1/4$, thereby answering Mubayi's question in these ranges.Comment: 15 page

Books versus triangles at the extremal density

A celebrated result of Mantel shows that every graph on n vertices with [n²/4] + 1 edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must, in fact, be at least [n/2] triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erdős, says that any such graph must have an edge which is contained in at least n/6 triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any 1/6 ≤ β 0 such that any graph on n vertices with at least [n²/4] +1 edges and book number at most βn contains at least (γ - o(1))n³ triangles. He also asked for a more precise estimate for γ in terms of β. We make a conjecture about this dependency and prove this conjecture for β = 1/6 and for 0.2495 ≤ β < 1/4, thereby answering Mubayi's question in these ranges

Short proofs of some extremal results III

We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short

Short proofs of some extremal results III

We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short

Books versus Triangles at the Extremal Density

A celebrated result of Mantel shows that every graph on n vertices with left perpendicularn(2)/4right perpendicular + 1 edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must, in fact, be at least left perpendicularn/2right perpendicular triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with ErdOs, says that any such graph must have an edge which is contained in at least n/6 triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any 1/6 0 such that any graph on n vertices with at least left perpendicularn(2)/4right perpendicular + 1 edges and book number at most beta n contains at least (gamma - o(1))n(3) triangles. He also asked for a more precise estimate for gamma in terms of beta. We make a conjecture about this dependency and prove this conjecture for beta = 1/6 and for 0.2495 <= beta < 1/4, thereby answering Mubayi's question in these ranges.ISSN:0895-4801ISSN:1095-714