An Exact Turán Result for Tripartite 3-Graphs

Mantel’s theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let K− 4 = {123, 124, 134}, F6 = {123, 124, 345, 156} and F = {K− 4 , F6}: for n 6= 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tripartite 3-graph S3(n) (for n = 5 it is C (3) 5 = {123, 234, 345, 145, 125}). This extends an old result of Bollobás that S3(n) is the unique 3-graph of maximum size with no copy of K − 4 = {123, 124, 134} or F5 = {123, 124, 345}.