This paper by P. Erdős explores extremal problems in graph theory, focusing on the structure of graphs that contain specific subgraphs. The main results include: 1. **Valence and Subgraph Existence**: For a graph \( G(n; m(n; p)) \), the smallest integer \( m(n; p) \) such that every such graph contains a complete graph \( K_p \) is determined. Additionally, it is shown that every \( G(n; m(n; p)) \) contains a \( K_{p+1} \) with one edge missing. 2. **\( K(p, p) \) Subgraph**: The smallest integer \( u(n; p) \) such that every \( G(n; u(n; p)) \) contains a \( K(p, p) \) is not known, but bounds are provided for \( u(n; 2) \). The limit of \( u(n; 2) / n^{3/2} \) is conjectured to be \( 1/2\sqrt{2} \). 3. **Refined Theorem 1**: A constant \( \gamma_{p} \) is introduced such that every \( G(n; [\gamma_{p} n^{2-1 / p}]) \) contains a \( K(p+1, p+1) \) with one edge missing. 4. **Theorem 2**: For any integer \( l > p \), a constant \( \gamma_{p, l} \) is found such that for \( n > n_0(p, l) \), every \( G(n; [\gamma_{p, l} n^{2-1 / p}]) \) contains a subgraph \( H(p, l, l) \), which is a \( K(l, l) \) with certain edges missing. The proofs involve several lemmas, including the existence of subgraphs with high valence and the construction of specific subgraphs within larger graphs. The results highlight the intricate structure and properties of extremal graphs in graph theory.This paper by P. Erdős explores extremal problems in graph theory, focusing on the structure of graphs that contain specific subgraphs. The main results include: 1. **Valence and Subgraph Existence**: For a graph \( G(n; m(n; p)) \), the smallest integer \( m(n; p) \) such that every such graph contains a complete graph \( K_p \) is determined. Additionally, it is shown that every \( G(n; m(n; p)) \) contains a \( K_{p+1} \) with one edge missing. 2. **\( K(p, p) \) Subgraph**: The smallest integer \( u(n; p) \) such that every \( G(n; u(n; p)) \) contains a \( K(p, p) \) is not known, but bounds are provided for \( u(n; 2) \). The limit of \( u(n; 2) / n^{3/2} \) is conjectured to be \( 1/2\sqrt{2} \). 3. **Refined Theorem 1**: A constant \( \gamma_{p} \) is introduced such that every \( G(n; [\gamma_{p} n^{2-1 / p}]) \) contains a \( K(p+1, p+1) \) with one edge missing. 4. **Theorem 2**: For any integer \( l > p \), a constant \( \gamma_{p, l} \) is found such that for \( n > n_0(p, l) \), every \( G(n; [\gamma_{p, l} n^{2-1 / p}]) \) contains a subgraph \( H(p, l, l) \), which is a \( K(l, l) \) with certain edges missing. The proofs involve several lemmas, including the existence of subgraphs with high valence and the construction of specific subgraphs within larger graphs. The results highlight the intricate structure and properties of extremal graphs in graph theory.