Extremal Graphs Without 4-Cycles

Sidon sets and graphs without 4-cycles

“…Along with the constructions of [4] and [7], the results of Füredi are the most important contributions to the 4-cycle Turán problem. Recently Firke, Kosek, Nash, and Williford [9] proved that for even q, ex(q 2 + q, C 4 ) ≤ 1 2 q(q + 1) 2 − q. If q is a power of two then we have the exact result ex(q 2 + q, C 4 ) = 1 2 q(q + 1) 2 − q.…”

“…Assume now that αc 1 + 2c 1 d 1 − β − d 2 1 = 0. Then (9) gives a unique solution for Y . Since (X, Y ) = (c 1 , d 1 ) is a solution we must have that all solutions to the system (7), (8) have Y = d 1 .…”

“…Then c 1 = 0 and d 1 = 0 otherwise c 1 = d 1 which we already know does not occur. Since both coefficients of X and Y are 0 in (9) the constant term must also be 0 so, using…”

Turán numbers of theta graphs

“…It is now known that for certain values of n the extremal graphs must come from projective planes [16,19,15] and this is conjectured to be the case for all n (see [17]).…”