On shifted intersecting fa:rr:tilies "VV'"ith r e s p e c t to p o s e t s Morimasa Tsuchiya Department of Mathematical Sciences Tokai University Hiratsuka, Kanagawa 259-12, JAPAN In this paper, we show that for a shifted complex :Jr Abstract. ~ 2P with respect to a poset P with minimum element 0 and an intersecting subfamily ~ ~ ~ #~ :Jr, #{FE:Jr;OEF}. We denote the set {1,2, .. ,n} by [n], the family of all subsets of a set X by 2x , #F denotes the number of elements of a set F. be a family of subsets of [n], i.e., E :Jr, Fi () F j '* = {F 1, .. ,F m } where F1 Fm A family g, is intersecting if for every F i , are distinct subsets of [n]. Fj :Jr Let :Jr For families ~, g, 0. '* cross-intersecting if G () F 0 for V G E family :Jr ~ 2[n] is called a complex if G S;;; F ~ and :Jr are S;;; 2[n], ~ and V F E E g, implies G :Jr. A E :Jr. We already know the following results. For an intersecting family :Jr S;;; 2[n J , #:Jr ~ 2n- 1 ([1]) and for a complex :Jr S;;; 2[n] and cross-intersecting subfamilies ~, Je ~ For F, G with x ~ ~ :Jr, #~ + #Je ~ # g, ([4]). [n], if there exists a one-to-one mapping I(x) for each x E F, then we write F Australasian Journal of Combinatorics ~ ( 1992 ~ G. I: :Jr L pp. 53-58 F --> G ~ 2[n] ~ ~ is V-hereditary if G FE implies G ;} E V.Chvatal introduced ;}. this notion and proved the next result. Theorem A (1974 [2]). Let;} ~ 2[n] be a V-hereditary family and ~ be an intersecting subfamily of ;}. Then # ~ ~ #{ FE;} ; 1 E F }. • H.Era extended the notion of V.Chvatal and also showed the following result. Let P be a finite ranked poset with the minimum element O. ~ For F, G x ~ P, if there exists a one-to-one mapping f: F --> G with f(x) in P or x and f(x) are incomparable for each x ~p we write F implies G E G. ;} ~ 2 P is P-hereditary if G E F, then ~p FE;} ;}. Let P be a finite ranked poset with the minimum Theorem B ([3]). ~ 2 P be a P-hereditary family. element 0 and ;} subfamily i:§ of:f, For an intersecting #~ ~#{FE;} ;OEF} . • Let P be a finite poset with the minimum element O. For a family ;} ~ 2P and a ~ (3 in P, we define S a,/3(F) = (F-{(3})u{a} if ai-F, (3EF,(F-{(3})U{a}i-;} { F for each F E ; } and Sa,/3(;}) =#:J and if ;} otherwise = {Sa,/3(F); FE:J}. is complex and intersecting, Then #Sa,/3(;}) then Sa, /3 (;}) is also complex and intersecting.' Proposition 1. complex, For a finite poset P and a, ~ (3 in P, if ;} ~ 2 P is then Sa, /3 (;}) is also complex. Proof. We suppose that there exist G, F Sa,/3(;}) and G i- Sa,/3(;})' Case 1. FE:;} 54 ~ P such that G ~ F E Since Y is complex, G ¢. Y and (3 F. E If E Y. a E ¢. G, (3 So a F, then (G-{(3})u{a} Y and ~ (G-{ (3}) u {a} ~ which F, If a ¢. F, then (F - { (3 }) u { a } contradicts the property that g; is complex. E G, (G-{ (3)) u { a} E (F -{ (3}) u {a}, which contradicts the property that g, is complex. Case 2. F ¢. g; . Then a a F, (3 E ¢. G, !3 E G' G and (G-{!3})u{a} If G contradiction. = (G-{a})u{!3} = G E Sa,(3(g;), ¢. Y, then g;. E Since G'n{a,!3} which is a contradiction. (F -{ a}) u {!3} and (G'-{!3})u{a} = {.B}, • ~ !3 in P, if g; ~ 2 P is Sa,(3(g;) such that GnF E a Thus there exists HEY such that Sa. (3 (H) =I F. By the definition of (a, !3)-shifting, H = (F-{ a}) u {(3} E F and !3 ¢. F. 0, which is a contradiction. Since ~ F, which is a Since Y is intersecting, both of G and F do not belong to Y. = F and H Y, i. Y, then E then Sa, (3 (Y) is also intersecting. We assume that F ¢. g;. E If G So G (G-{ a}) u {!3} Proof. We suppose that there exist G , F = O. Y. E ¢. Y. For a finite poset P and a Proposition 2. intersecting, ¢. F and (F-{a})U{!3} Sa, (3(G) ( a, !3 )-shifting. Then «F-{ a })n(G-{ (3}» ({a}n{(3}) Thus G u =(F-{a})n(G-{!3}) = 0, g; n ({(3}n(G-{!3}» E E by G and F n G G and the a =I ¢. G. definition of «G-{!3})u{a}) u «F -{ a}) n { a }) u contradicting the fact that Y • A family g; is shifted if Sa,(3(Y) P. E «F-{a))u{!3}) is an intersecting family. < (3 in :J, !3 E (G-{(3})u{a} G, = ¢. :J, then a If G = g; for all a, !3 such that a We obtain the following result which is concerned with shifted complexes and intersecting families. 55 Let P be a finite poset with the minimum element 0 and Theorem 3. S 2 S;;; #~ ~ P For an intersecting subfamily ~ of S, be a shifted complex. #{ FES; OEF}. = Proof. Let :reO) Proposition 2, we can assume that ~... family H E ~}, = {H ; H S;;; :3 G ~.... E = { FE S {F-{O}; OEFE fJ } and So ~ is shifted. that is, if G In the following we show that ~... By Then we define the ~ E ; OiF}. and H G, S;;; then = {H ; H s;;: :3 G E ~} is a shifted complex. Suppose that ~.. is not a shifted complex. and P E H (H- {/3})v{a} that H ~ is i shifted. ~ ... a ;;; (H - {/3}) V { a} E H n { a, /3 } /3 , = {/3}, then (G - (H-{/3})V{a} {/3}) V { a} ~ .. , S;;; E H :3G E E ~} is a shifted complex and ~ ~. Since ~ .. is shifted, ~... Let ~ 0 == { G E ~ ; 0 i G } and '{5 ~ For C 0. #~ 'f3 and G E For '{5+ ~o' since 0 By the fact that ~ 0'. = = E ~ #~o. i #S(O). == i 0 = { C E fJ 0; Since ~o and So- 'f3 are cross-intersecting, and therefore #'f3 S;;; g;. #~ - #~o + #'{5 Proposition 4. ::/. {/3}, G E ~, == = c i 0, #'f3+ 0'}. ~ =#'{5. ({O}vC)nG ~. #S(O). So 'B + n ~ 'f3 + S;;; S, • Let P be a finite poset with the minimum element O. 56 E + #(So- '(3) ;;;;;; #fJ o G and enG #«~ - ~o)U '(5+) S;;; Therefore :3 G E ~ 0' C n G == #~o {O} V ~.. H implies RU{O} ={CV{O};C E 'f3}, is intersecting, S;;; For Y H Since every element of (~- ~ 0) U '{5 + contains 0 and ;;;;;; ~, E which is a contradiction. = {H ; H S;;; :3 G c ~ because E (G - {/3}) V {a} without loss of generality we can assume that ~.. ~ .. -~, and ~ such If G n { a , /3} So for ~ .. (O) = {G-{O}; OEGE~ .. }, #~ .. (O) S. /3 { /3} (G- {/3})v{a} S;;; ~ .. , which is a contradiction. Since (H - {/3}) V {a} G. E Thus ~... that By definition of ~ .. , there exists G E Since (H - { /3 }) V { a} E then a, /3 such If G n { a ,/3} G. S;;; ~.. E Then there exist a, If g. , 2 P is a P-hereditary family, Proof. We assume that G , 2 P then g. is a shifted complex. and G , J F E: g.. Since the mapping ~p f from G to F such that f(x) = x is a one-to-one mapping, G By the property that g. is a P- hereditary family, G g.. E: F. Thus g. is complex. that a .f3 E: P and a and (F-{/3})V{a} i /3 and ~ g.. /3 such F n { a, /3} = {/3} Then there exist a and We assume that g. is not shifted. F E: g. such that We define the mapping f from (F-{/3})V{a} to F as follows: f(x) = Since a if x -=I a X { ~ /3 if x = a. /3 in P, x ~ f(x) for \Ix E: ~p a one-to-one mapping and (F-{/3})v{a} g. is a P-hereditary family, (F-{ /3}) u {a} E: Thus f is (F-{/3})v{a}. F. By the property that g., which is a contradiction . • By Proposition 4 and Theorem 3, we also obtain Theorem B. the converse of Proposition 4 does not hold. = of Figure 1, :J shifted complex. P-hereditary. However For example, for the poset {{O,1,2}, {O,l}, {O,2}, {1,2}, {a}, {I}, {2}} is a Since {3} ~P {2} and {3} i g., g. is not So we do not obtain Theorem 3 from Theorem B. We can easily see that ,Cj is a V-hereditary family if and only if g. is a shifted family with respect to a linear order set. Let P be a poset with the minimum element and I(P) be a liner extension of P. If g. is a shifted family with respect to I(P), then g. is a shifted family with respect to P. So we also obtain Theorem; A by Theorem 3. converse does not hold. For example, :J = {{O,1,2}, But the {O,3,4}} is a shifted family with respect to the poset of Figure 1 and is not a shifted family with respect to the liner extension 57 ° ~ 1 ~ 2 ~ 3 ~ 4. Figure 1. References. [1] I.,Anderson, Combinatorics of finite sets,Oxford Univ. Press, (1987). [2] V.,Chvatal, Intersecting families oj edges in hyper graphs having the heredita ry property ,Hypergraph Seminar, LNM 411,Springer (1974) 61-66. [3] H.,Era, A comment on a Chvatal's conjecture, [4] L,Marica and J.,Schonheim, preprint. Differences of sets and a problem of Graham, Can. Math. Bull. 12 (1969) 635-637. 58