Ex(n;4)

ex(n;4)
P.Erd&ouml;s conjectured that ex(n;4)=(1/2 + o(1))3/2 n3/2.

Garnick et al. demonstrated that extremal graphs include:
 * Stars K1,n-1 and Paths Pn for n ≤  4,
 * Moore graphs:
 * EX(5;4)={C5} the cycle;
 * ex(10;4)= 15 is attained by the Petersen graph ;
 * ex(50;4)=175 is attained by the Hoffman-Singleton graph;
 * ex(3250;4) is attained by the Moore graph with girth 5 and degree 57 if it exists.
 * C6 is an element of EX(6;4).
 * polarity graphs.

Large graphs, constructed by researches interested in The Cage Problem, provide good constructive lower bounds for the extremal number. These graphs are listed in the table when the number of vertices is less than 204. The higher order graphs produce the following lower bounds. exl(240;4)=1560, exl(288;4)=2016, exl(312;4)=2340,exl(336;4)=2688, exl(448;4)=3808, exl(480;4)=4320,exl(512;4)=4864 and exl(576;4)=5760.

Abajo, Balbuena and Di ́anez give the following constructive lower bounds for the extremal number:
 * exl(2q2+q;4)=q2(q+1)+(q+1)ex(q;4) where q>1 is a prime power.
 * exl(2q2+q-h;4)=q2(q+1)+(q+1)ex(q;4)-hq+e where q=8,9,11, e=-h for h=1,2 and e=-2 for h=3,...,q+1.

The following table lists the known values and constructive lower bounds for ex(n;4) for n < 204. If the value is known then it is shown in bold text.

The following table is the key to the colors in the table presented above: