An extremal problem concerning graphs not containing \(K_t\) and \(K_{t,t,t}\) (Q2576851)

The authors prove the following results: Let \(t\) be a positive integer with \(t \geqslant 4\). Then the maximum number of edges of a graph of order \(3 t\) that contains neither \(K_t\) nor \(K_{t,t,t}\) equals \[ \binom {3 t}{2} - 3t - 7 \quad\text{ if } \quad t \equiv 1\bmod 3,\tag{1} \] \[ \binom {3 t}{2} - 3t - 6 \quad\text{ if } \quad t \geqslant 2\bmod 3,\tag{2} \] and equals \[ \binom {3 t}{2} - 3t - \frac {t}{3} -3 \quad\text{ if } \quad t\equiv 0\bmod 3.\tag{3} \] If \(t \equiv 1\bmod 3\), then the only graphs of order \(3 t\) that contain neither \(K_t\) nor \(K_{t,t,t}\) as a subgraph and whose number of edges equals (1) are the graphs of the forms \(K_{3, \dots,3,4,5}\), \(K_{3,\dots,3,4}+\bar {A}\) where \(A\) is a join of two \(K_{4}\)'s, and \(K_{3,4,4,4}+\bar {B}\) where \(B\) is a join of two \(K_{3}\)'s or \(K_{2,3,4,4,4,4}\) when \(t = 7\).