A,, be finite sers such that A,@ A, for all i * j. Let F be an intencctlng family con&ting of sets contained in some A,. i = 1. 2. . . . n. I_'hvital conjecl urtxl that among the largest irtersecting families. there is always a star. In Ihi\ pi per. we oNam another proof of a result of Schiinheim: If A, fl A, fl --. n A, f QI. the11 the conjecture is trul:. WC alw pmvc' that ;f P., fI A, n A,, = Q! for all i # j# k-f i or if the independent iwtern \atisfies a hercditar, trc'c strut tuw. then the coniecturc is also true.
4). IntYobction
A finite family F of finite sets is called an independent system if X E F, YC X3 YE F. A family of sets is cailed intersecting if it contallls no two dis.,oint sets. It is called a star if all of its acts have at leasl: one element in common. For example, the family 8, {I), {2j, (31, {1,2), { 1,3}, {2,3) is an independent system; its subfamily {I}, {1,2j. (I, 3) is a star, the subfamily (1,2). { 1,3}. (2, 3) is intersectin;? but not a star. Chvfital conjectured that among the largest intersecting subfamilk of an independent system, there is always a s!ar. In this paper, we prolIe some special cases of the conjecture.
Let A,, AZ,. . .9 A, be the maximal sets in the indeper,dent systeni I' (i.e. A, E B.. B E I~Ai=61).WeshowthatifA,nA,n**.nA"#~otifAink,17A,=p)forall i# i# k# i, then the conjecture is true. We also obtain some generalizations about the hereditary families [ 11.
1,
In this section, we give another prooi of the io!!oky resrilt of Scnijnheim [ 5 1:
Theorem 1.1. If A, n Al n -l f~ A,, # fl. rhen the cmjectu-e is trut .
r irst vie gille some definitions. Two collt ctions F, ;Ind r, ;,1' 5;lik sets ark-2 cdilcd . _ intersectin; II' each set in F, has a non-empty interzectior with evq WY irl I-,.