On Turán densities of small triple graphs

For a family of kk-graphs FF, the Turán number T(n,F)T(n,F) is the maximum number of edges in a kk-graph of order nn that does not contain any member of FF. The Turán density t(F)=limn→∞T(n,F)/nk. Let K4K4 be the tetrahedron that is the complete triple graph of order four. Let the triple graph Fp,qFp,q defined on the vertex set P∪QP∪Q with |P|=p|P|=p and |Q|=q|Q|=q consist of those edges which intersect PP in either one or three vertices. Let V={1,2,3,4,5}V={1,2,3,4,5} and let F5F5 be defined on VV with E(F5)={123,145,245}E(F5)={123,145,245}. Let F5c denote the complement of F5F5, and let P5P5 be the weak pentagon obtained from F5F5 by adding the edge 134134 and let C5C5 be the pentagon obtained from P5P5 by adding one more edge 235235. We prove that•1/2≤t(F1,4)≤2/31/2≤t(F1,4)≤2/3,•23−3≤t(K4,F5c)≤2−2,•1/4≤t(F1,4,P5)≤62−8,•2/9≤t(F1,3,F3,2,C5)≤1/11. The first result relates to a conjecture of Mubayi and Markström–Talbot that t(F1,3)=2/7t(F1,3)=2/7. The best known bounds are 2/7≤t(F1,3)<0.329082/7≤t(F1,3)<0.32908. The second result relates to an old conjecture of Turán that t(K4)=5/9t(K4)=5/9. The best bounds are 5/9≤t(K4)≤(3+17)/12. The last two results relate to a result of Mubayi and Rödl that t(F1,3,C5)≤10/31t(F1,3,C5)≤10/31.