By Isac G.

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**Example text**

Clearly F 1 contains every t-set not involving ~, F 3 contains every t-set involving =. Since F 1 + F 3 may not be as economical as possible, we have Theorem 2. 3. N(t,k,v) ,< N(t,k,v-l)+N(t-l,k-l,v-l). KNOWN RESULTS An extensive survey of results appears in [3]; a very excellent and up-to- date survey with many important new results appears in [2]. Briefly, the following results are known (for complete references, see the extensive reference list in [2]). L+2 (I) N(2,3,v) = L(2,3,v), a result proved by Fort and Hedlund.

2) N(2,4,v) = L(2,4,v) for v ~ 7,9,10,19 (the result is L + 1 for 7,9,10 ; for 19). (3) N(2,5,v) has been studied by Gardner and others; (4) N(k-l,k,v) has been studied; some results are known. many results are known, but the only com- plete determination is for N(3,4,v), with v ~ 7, mod 12. given by the bound L(3,4,v). In those cases, N(3,4,v) is For v _-- 7 rood 12, the only known result is that N(3,4,7) = 12, whereas the bound is ii. (5) Numerous Steiner Systems S(t,k,v) are known; ting appears in [2].

V I f n >~ ~ - a n d m > ( t - 1 ) n , = Ik_t+I l . than Form a d e s i g n on n b l o c k s w h i c h c o n t a i n s varieties; " then B ( t , k , m , v ) g n. an m - s e t w h i c h i s n o t c o v e r e d by t h i s e a c h b l o c k i n a t most t - 1 v-m k-t+l each variety design, it so we would h a v e m < n ( t - 1 ) . at least once. would h a v e t o meet This is a contra- diction, Lemma 3 . 9 . Proof. B(2,k,m,v) If I ~ I ¢ m. 8 shows that B(2,k,msV) < m. it If can be altered to produce a design containing every variety.

### 0 - Epi families of mappings, topological degree, and optimization by Isac G.

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