By virtue of Proposition 3 and the conditions of our theorem, this graph contains a complete subgraph on n -- p + q vertices; let, for definiteness, these be the vertices {xI .... , zn_p+q}. Then, by the triangle inequality, we have [ ~n ~n--P-iq n ~.n , _ _ The construction of n -- p + q pairwise orthogonal vectors, supplemented by a bundle of p -- q vectors, in the direction of the vector of the sum of n- p + q vectors, realizes this esti- mate. LITERATURE CITED i. B. S. Stechkin, "Generalized valences," Mat. Zametki, 17, No. 3, 433-442 (1975). 2. P. ErdDs and J. Spencer, Probabilistic Methods in Combinatorics, Academic Press, London-- New York (1974). 3. J. Suranyi, "Kombinatorikus geometriai problemak," Mat. Lapok, 22, Nos. 3-4, 215-230 (1971). 4. G. Katona, "A theorem of finite sets," in: Theory of Graphs, Akad~miai Kiad6, Budapest (1968), pp. 187-207. 5. P. Erd~s, "A problem on independent r-tuples," Ann. Univ. Sci. Budapest, 8, 93-95 (1965). 6. P. Erd~s, "On a lemma of Hajnal--Folkman," in: Combinatorial Theory and Its Applications Akad~miai Kiad6, Budapest (1970), pp. 311-316. 7. G. N. Kopylov, "Extremal (p.q) graphs," in: Fourth All-Union Conference on the Prob- lems of the Theory of Cybernetics [in Russian], Abstracts of Reports, Novosibirsk (1977), pp. 138-139. 8. G. N. Kopylov, "Generalization of the Turin theorem," Mat. Zametki, 26, No. 4, 593-602 (1979). 9. B. S. Stechkin and P. Frankl, "The local-Tur~n property for ~-graphs," Preprint No. 20, Mathematics Institute, Academy of Sciences of Hungarian Peoples Republic, Budapest (1977). i0. D. Katona and B. S. Stechkin, "Combinatorial numbers, geometric constants, and probabil- istic inequalities," Dokl. Akad. Nauk SSSR, 251, No. 6, 1293-1296 (1980). ii. A. F. Sidorenko, "Classes of hypergraphs and probabilistic inequalities," Dokl. Akad. Nauk SSSR, 254, No. 3, 540-543 (1980). RELATIONS BETWEEN BEST APPROXIMATIONS OF FUNCTIONS OF TWO VARIABLES V. N. Temlyakov Let X denote either the space C or the space L of functions on the two-dimensional torus, En, m (/)x be the best approximation to the function f(x, y) by trigonometric polynom- ials of order n in x and order m in y in the metric of X, En,~ (/)x be the best approximation by trigonometric polynomials of order n in x with coefficients depending on y; analogously, one defines E~,m (/)x. Bernshtein [i] posed the problem of estimating the best approximations E~.m (/)c in terms of En,~ (/)c and E~.m ~)c- He proved the inequality En. m (/)c < A In {2 + rain (n, m)} [En ~ q)c + E~. m q)c], (i) where A is an absolute constant. Later it was proved that a similar estimate is true also for approximations in the metric of L (see [2]). It is proved in [3] that in (i) and in the similar relation for the metric L the factor in {2 + min (n, m)} cannot be changed to any other factor, growing slower as min (n, m)--~ than In {2 +min (n,m)}. In particular (m = n) the following assertion is proved [3, Theorems 3 and 4]. V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR. Translated from Matematicheskie Zametki, Vol. 29, No. i, pp. 95-106, January, 1981. Original article sub- mitted July 5, 1978. 0001-4346/81/2912-0051507.50 9 1981 Plenum Publishing Corporation 51