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Statement

Notation as on the Theorem 1 page: $f(n,H)$ is the largest number of colors on the edges of $K^n$ with no totally multicolored (TMC) copy of $H$, and $d+1=\min\{k(H-e):e\in E(H)\}$, $k$ the chromatic number.

Theorem 3 (printed p. 635, quoted). "Let us consider a $K^n$ coloured by $f(n,H)$ colours and not containing TMC $H$. One can subdivide $V(K^n)$ into $d$ sets $A_1,\ldots,A_d$ (where $d$ was defined in Theorem 1) so that all but $o(n^2)$ edges joining different classes have own colours (i.e., colours used only once) and all the edges joining vertices of the same class $A_j$ have $o(n^2)$ colours altogether, $(j=1,2,\ldots,d)$."

The $o(n^2)$ terms are as $n\to\infty$ with $H$ fixed.

Source. P. Erdős, M. Simonovits and V. T. Sós, Anti-Ramsey theorems, Infinite and finite sets (Colloq., Keszthely, 1973), Vol. II, Colloq. Math. Soc. János Bolyai 10, North-Holland (1975), 633–643; printed p. 635. The edition is identified in the source digest.

Read depth. Claims checked: the statement was read clause by clause on the page image. The paper prints no proof.

Proof pointer

None in the paper: "The proof of this theorem will not be published here" (p. 635). The paper says it can be proved by using results of Erdős (Some recent result in extremal graph problems in graph theory, Theory of Graphs, Proc. Symp. Rome (1966), 118--123) and Simonovits (A method for solving extremal problems in graph theory, stability problems, Theory of Graphs, Proc. Symp. Tihany (1966), Acad. Press (1968), 279--319) in place of the Erdős--Simonovits limit theorem, its (1) on p. 634, and combining the method of the proofs of Theorems 1 and 2 with that of Theorem 4.

Dependencies

None in the paper.

Bears on

No problem page of this corpus.