# Large triangle-free subgraphs in graphs without*K*_{4}

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## Abstract

It is shown that for arbitrary positive*ε* there exists a graph without*K*_{4} and so that all its subgraphs containing more than 1/2 +*ε* portion of its edges contain a triangle (Theorem 2). This solves a problem of Erdös and Nešetřil. On the other hand it is proved that such graphs have necessarily low edge density (Theorem 4).

Theorem 3 which is needed for the proof of Theorem 2 is an analog of Goodman's theorem [8], it shows that random graphs behave in some respect as sparse complete graphs.

Theorem 5 shows the existence of a graph on less than 10^{12} vertices, without*K*_{4} and which is edge-Ramsey for triangles.

## Keywords

Random Graph Complete Graph Edge Density
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