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let @xmath11 be an integer and @xmath4:=\{1,2,\dots , n\}$ ] . by @xmath12}$ ] we denote the family of all subsets of @xmath4 $ ] and by @xmath13}k$ ] the family of all @xmath14-subsets of @xmath4 $ ] . a family @xmath15}$ ] is an _ antichain _ if @xmath5 for all distinct @xmath16 . an antichain @xmath17 is called _ flat _ if it is contained in two consecutive levels of @xmath12}$ ] , i.e. if @xmath18}{k}\cup\tbinom{[n]}{k+1}$ ] for some @xmath14 . more generally , for a subset @xmath19 $ ] , we call @xmath17 a _ @xmath1-antichain _ if it contains only @xmath14-sets with @xmath20 , i.e. @xmath21 where @xmath22}{k}$ ] . the _ dual _ @xmath23 of a family of sets @xmath24 is the collection @xmath25 of subsets of @xmath26 $ ] given by @xmath27\ : \ i \in a_j\}$ ] for @xmath28 $ ] . it is well known that a family of sets @xmath17 is an antichain if and only if its dual @xmath23 is a _ completely separating system _ ( css ) : a css is a collection @xmath29 of blocks of @xmath4 $ ] such that for each @xmath30 $ ] there are blocks @xmath31 with @xmath32 and @xmath33 . the dual of a flat antichain on @xmath13}{k } \cup \binom{[n]}{k-1}$ ] is a _ fair css _ : a css in which each point occurs exactly @xmath14 or @xmath34 times . the consideration of minimum size csss led to a conjecture which subsequently became the _ flat antichain theorem _ , which follows from results of lieby @xcite ( see also @xcite ) and kisvlcsey @xcite . this theorem greatly reduced the computational search space for csss . it says that for every antichain @xmath17 there is an equivalent flat antichain @xmath35 , where the equivalence relation is defined by : @xmath35 is equivalent to @xmath17 if and only if @xmath36 and @xmath37 . griggs et al . @xcite showed that the flat antichains minimize the _ blym - values _ @xmath38 within their equivalence classes , and more generally , they minimize ( maximize ) @xmath39 for every convex ( concave ) function @xmath40 . csss have applications in various areas including coding theory , search theory , and topology . see  @xcite and  @xcite for applications and further references for csss . a @xmath1-antichain @xmath17 is called a _ maximal @xmath1-antichain _ if there is no @xmath1-antichain @xmath35 with @xmath41 . for @xmath42 , any @xmath43-antichain is also a @xmath1-antichain , and if it is a maximal @xmath1-antichain , then it is also a maximal @xmath43-antichain . the converse is not true as can be seen by the family @xmath44 for @xmath45 , which is a maximal @xmath46-antichain but not a maximal @xmath47-antichain , since @xmath48 is an antichain properly containing @xmath17 . this example is illustrated in figure  [ fig : maximality ] , showing a graph on 9 vertices whose edge set is the complement of @xmath49 in @xmath50}{2}$ ] , while the set of 4-cliques is precisely @xmath51 . -antichain which is not maximal as a @xmath47-antichain , and thus also not strongly maximal . ] we also call a @xmath1-antichain @xmath17 _ strongly maximal _ if it is maximal as an antichain , not just as @xmath1-antichain , i.e. if there is no antichain @xmath35 with @xmath41 . equivalently , @xmath17 is a strongly maximal @xmath1-antichain if it is also a maximal @xmath52-antichain , where @xmath14 and @xmath53 are the smallest and the largest element of @xmath1 , respectively . in this paper , we always assume @xmath54 and @xmath55 , and we study the problem of determining the smallest possible cardinality of a maximal @xmath1-antichain . our approach to the minimum size of a maximal antichain is based on the graph interpretation which was used in  @xcite to completely resolve the problem for the case @xmath56 . to prove our asymptotic results we adapt arguments from  @xcite where lower bounds for the size of maximal @xmath57-antichains were proved . the main tool in this part is the graph removal lemma . let @xmath58 be a graph on @xmath59 vertices and let @xmath60 . then there exists @xmath61 such that every graph @xmath62 on @xmath63 vertices containing at most @xmath64 copies of @xmath58 can be made @xmath58-free by removing @xmath65 edges . this was proved , using szemerdis regularity lemma , first for @xmath58 being a triangle  @xcite and later in general  @xcite . see  @xcite for a recent proof avoiding the regularity lemma , and thus improving the bounds on @xmath66 significantly . the paper is organized as follows . in section  [ sec : graph ] the problem is reformulated as a question in extremal graph theory . in section  [ sec : construction ] we describe a construction for graphs , which we believe to correspond to optimal antichains . some evidence for this conjecture is provided by exhaustive search results for @xmath10 and small @xmath63 which are presented in section  [ sec : small_n ] . section  [ sec : asymptotics ] contains our asymptotic results : if @xmath7 our construction yields the correct leading term , and for @xmath10 we prove a first nontrivial bound . finally , in section [ sec : problems ] we suggest some open problems for further investigations . as proposed in  @xcite , we associate a graph @xmath67 with an antichain @xmath17 . the vertex set is @xmath68 $ ] and the edge set @xmath69}{2}\setminus\mathcal a_2.\ ] ] we start by making some simple observations . [ lem : simple_observations ] let @xmath17 be a @xmath1-antichain on @xmath63 points and let @xmath70 be the associated graph .    1 . every @xmath71 induces a clique in @xmath62 . if @xmath17 is a maximal @xmath1-antichain and @xmath72 is the vertex set of a @xmath14-clique in @xmath62 for some @xmath20 then there is an @xmath73 with @xmath74 or @xmath75 . in particular , every edge of @xmath62 is contained in a @xmath14-clique for some @xmath76 . the first statement is just the antichain property applied to @xmath77}{2}\setminus e$ ] and @xmath71 . the second statement follows from maximality , because if no such @xmath78 exists then we can add @xmath72 to @xmath17 to obtain a strictly larger antichain . the elements of the antichain are the non - edges of the graph and some cliques whose sizes are in @xmath79 . we say that a graph is _ @xmath1-saturated _ if every edge is contained in a @xmath14-clique for some @xmath76 . being @xmath1-saturated is a necessary and sufficient condition for a graph to be @xmath80 for some maximal @xmath1-antichain @xmath17 . in general , it is not possible to reconstruct the maximal @xmath1-antichain uniquely from the graph : if @xmath81 then a 4-clique can ( among others ) come from a single 4-set or from four 3-sets in @xmath17 . for a @xmath1-saturated graph @xmath70 with @xmath68 $ ] let @xmath82 denote the set of all @xmath1-maximal antichains @xmath17 with @xmath83 . in other words @xmath82 is the collection of all families @xmath84}$ ] such that    1 . @xmath85}{2}\setminus e$ ] , 2 .   for all @xmath76 every @xmath86 induces a @xmath14-clique in @xmath62 , and 3 .   for all @xmath76 and every @xmath14-clique @xmath72 in @xmath62 , there is a @xmath87 and an @xmath88 with @xmath74 or @xmath89 .    a greedy choice among all the maximal @xmath1-antichains corresponding to a given @xmath1-saturated graph @xmath62 is based on the intuition to take the sets in @xmath17 as large as possible for the given graph @xmath62 . more precisely , for @xmath20 , let @xmath90 denote the set of @xmath14-cliques in @xmath62 that are not contained in a @xmath91-clique for any @xmath87 with @xmath92 . note that for @xmath1-saturated graphs @xmath93 . we put @xmath94 and define the maximal @xmath1-antichain @xmath95 by @xmath96}{2}\setminus e\right)\cup\mathcal m.\ ] ] it is not necessarily the smallest maximal @xmath1-antichain corresponding to the graph @xmath62 : for @xmath97 and @xmath62 the complete graph on @xmath63 vertices we obtain @xmath98}{4}$ ] , while @xmath99}{3}$ ] is a smaller maximal @xmath1-antichain corresponding to the same graph . but under the following sparseness condition on @xmath62 the antichain @xmath100 has minimum size . let @xmath101 be the minimal element of @xmath79 . we say that a @xmath1-saturated graph is _ @xmath1-sparse _ if for every @xmath102 , there exists a @xmath101-clique @xmath103 such that @xmath72 is the unique element of @xmath104 containing @xmath105 . the next lemma asserts that in this situation the antichain @xmath100 has minimum size among all @xmath1-antichains with corresponding graph @xmath62 . [ lem : canonical_ac ] let @xmath70 be a @xmath1-sparse , @xmath1-saturated graph , and let @xmath106 . then @xmath107 for all maximal @xmath1-antichains @xmath108 . let @xmath35 be any maximal @xmath1-antichain with @xmath109 . clearly @xmath110}{2}\setminus e$ ] . let @xmath111 be the set of @xmath101-cliques in @xmath62 that are contained in exactly one element of @xmath104 . since @xmath62 is @xmath1-sparse there is a mapping @xmath112 with @xmath113 for every @xmath102 . by the definition of @xmath111 , the mapping @xmath114 is injective . by the second part of lemma [ lem : simple_observations ] every element of the image of @xmath114 is contained in some member of @xmath35 . if the sets @xmath115 and @xmath116 are contained in the same @xmath117 , then @xmath118 for @xmath119 because @xmath120 is contained in a unique element of @xmath104 . this , in turn , implies @xmath121 for the same reason . hence @xmath122 is a lower bound for @xmath123 , and this proves the statement . the following lemma provides a characterization of the @xmath1-saturated graphs @xmath62 such that @xmath100 is a strongly maximal @xmath1-antichain . [ lem : strong_maximality ] let @xmath62 be a @xmath1-saturated graph for @xmath124 . then @xmath106 is strongly maximal if and only if the following holds . for every @xmath125 , every @xmath126 and every @xmath127-clique @xmath105 in @xmath62 , we have that either    1 .   [ cond_1 ] @xmath105 contains a @xmath128-clique that is not contained in any @xmath129-clique , or 2 . [ cond_2 ] @xmath105 is contained in a @xmath129-clique . suppose @xmath130 satisfies the condition and @xmath17 is not strongly maximal . then there is a set @xmath131 that can be added such that @xmath132 is still an antichain . the set @xmath105 must induce a clique in @xmath62 and by lemma [ lem : simple_observations ] this implies @xmath133 and @xmath134 . hence @xmath135 for some @xmath136 . but then either of the conditions [ cond_1 ] ( with lemma [ lem : simple_observations ] ) and [ cond_2 ] ( using the definition of @xmath100 ) implies that @xmath105 can not be added without violating the antichain condition . for the converse , suppose @xmath62 is a @xmath1-saturated graph containing a @xmath127-clique @xmath105 , @xmath137 , such that @xmath105 is not contained in any @xmath129-clique but every @xmath138-subset of @xmath105 is contained in a @xmath129-clique . by construction of @xmath106 , for every @xmath128-subset @xmath139 , the antichain @xmath17 contains a @xmath140-set @xmath141 for some @xmath142 . but this implies that @xmath132 is still an antichain , hence @xmath17 is not strongly maximal . to summarize , we have proved the following .    1 .   for every maximal @xmath1-antichain @xmath17 there is a unique @xmath1-saturated graph @xmath80 . 2 .   minimizing the size of a maximal @xmath1-antichain is equivalent to @xmath143 where the maximum is over the set of all @xmath1-saturated graphs @xmath62 and the minimum is over the set @xmath144 . if the graph @xmath62 is @xmath1-sparse then the antichain @xmath106 defined by ( [ eq : canonical_ac ] ) achieves the minimum in ( [ eq : reformulation ] ) . in this section , we describe a family of @xmath1-saturated graphs with many edges and few cliques ( corresponding to small maximal @xmath1-antichains ) . our construction is inspired by the optimal graphs for @xmath145-antichains in @xcite : we start with a complete bipartite graph with almost equal parts and add a few edges to make the graph @xmath1-saturated without producing too many cliques with sizes in @xmath1 . let @xmath53 be the maximal element of @xmath1 . the vertex set is partitioned into three disjoint sets @xmath72 , @xmath105 , @xmath146 with @xmath147 and the edge set is determined as follows .    * the induced subgraph on @xmath72 is the disjoint union of @xmath148-cliques . * the induced subgraph on @xmath105 is the disjoint union of @xmath149-cliques . * every pair @xmath150 with @xmath151 and @xmath152 is an edge . * the vertices in @xmath146 are isolated .    clearly every clique of at least two vertices is contained in an @xmath53-clique , so @xmath104 is the set of all @xmath53-cliques . any @xmath53-clique contains vertices from @xmath72 and vertices from @xmath105 , and every @xmath101-clique that has nonempty intersection with @xmath72 and with @xmath105 is contained in a unique @xmath53-clique . thus the graph @xmath62 is @xmath1-sparse and using lemma [ lem : canonical_ac ] the antichain @xmath100 , defined by ( [ eq : canonical_ac ] ) , has minimum size among the antichains in @xmath82 and contains only @xmath153- and @xmath53-sets . we get @xmath154 and @xmath155 thus @xmath156 the corresponding maximal @xmath1-antichain has size @xmath157 note that in this construction every @xmath127-clique for @xmath158 is contained in an @xmath53-clique , thus by lemma [ lem : strong_maximality ] the antichains @xmath100 are strongly @xmath1-maximal .      in the above construction we wasted the vertices in the set @xmath146 , and the solution can be slightly improved by connecting @xmath146 to the rest of the graph . the best way to do this depends on the remainder of @xmath63 modulo @xmath53 . here we describe what to do for @xmath159 and @xmath160 ( @xmath161 ) . : :    in this case @xmath163 $ ] , @xmath164 $ ] and    @xmath165 and the edge set is    @xmath166,\,j\in[2m+1,4m]\right\}\cup\{\{2i-1,2i\}\ : \ i\in[2m]\},\ ] ]    and we have    @xmath167 @xmath168 . : :    in this case @xmath163 $ ] , @xmath164 $ ] and    @xmath169 and the edge set is    @xmath170,\,j\in[2m+1,4m+1]\right\}\cup\left\{\{2i-1,2i\}\ : \ i\in[2m]\right\}\\ \cup\{\{4m,4m+1\}\},\end{gathered}\ ] ]    and we have    @xmath171 @xmath172 . : :    in this case @xmath163 $ ] , @xmath173 $ ] and    @xmath165 and the edge set is    @xmath166,\,j\in[2m+1,4m+2]\right\}\cup\{\{2i-1,2i\}\ : \ i\in[2m+1]\},\ ] ]    and we have    @xmath174 @xmath175 . : :    in this case @xmath163 $ ] , @xmath173 $ ] and    @xmath176 and the edge set is    @xmath170\cup\{4m+3\},\,j\in[2m+1,4m+2]\right\}\cup\{\{2i-1,2i\}\ : \ i\in[2m+1]\}\\ \cup\left\{\{2m,4m+3\}\right\},\end{gathered}\ ] ]    and we have    @xmath177    the construction is illustrated in figure [ fig : examples ] . -saturated graphs for different values of @xmath178 . all pairs of vertices on different sides of the dashed line are joined by an edge , but these edges are omitted in the pictures . ]    -saturated graphs for different values of @xmath178 . all pairs of vertices on different sides of the dashed line are joined by an edge , but these edges are omitted in the pictures . ]    -saturated graphs for different values of @xmath178 . all pairs of vertices on different sides of the dashed line are joined by an edge , but these edges are omitted in the pictures . ]    -saturated graphs for different values of @xmath178 . all pairs of vertices on different sides of the dashed line are joined by an edge , but these edges are omitted in the pictures . ] we conjecture that for large @xmath63 these graphs are best possible . [ conj : graph_conjecture ] if @xmath63 is large , in any @xmath46-saturated graph @xmath62 on @xmath63 vertices , we have @xmath179    in terms of antichains the equivalent statement is that for any maximal @xmath46-antichain @xmath17 on @xmath63 points , we have @xmath180 in this section we present some computational results for @xmath181 and @xmath182 . these results , summarized in table [ tab : small_cases ] , were obtained by solving binary programs for maximizing @xmath183 and @xmath184 , respectively , over all pairs @xmath185 of a @xmath1-saturated graph @xmath62 on @xmath63 vertices and an antichain @xmath186 . the binary programs were solved with the commercial mip solver cplex . .computational results for small values of @xmath63 . we report the size of the optimal antichain @xmath17 that was found and its profile vector @xmath187 where @xmath188 is the number of @xmath14-sets in @xmath17 . the last column contains the size of the antichain corresponding to the graphs constructed in section [ subsec : l_4 ] . [ cols= " > , > , > , > , > , > " , ]     we make the following observations .    1 .   for @xmath182 , the construction in section [ subsec : l_4 ] yields optimal @xmath47-saturated graphs . so the bound in conjecture [ conj : graph_conjecture ] might be true for @xmath47-graphs even without the restriction to large @xmath63 . 2 .   if @xmath189 , i.e. when the construction in section [ subsec : l_4 ] yields a regular graph , it is also optimal for @xmath46-saturation . 3 .   for @xmath97 all optimal examples that have been found do not contain any 3-set . a particularly interesting case is @xmath190 . the unique optimal @xmath46-saturated graph has the set of 4-cliques @xmath191 defining a @xmath192-regular graph in which every vertex is contained in exactly two 4-cliques , and every edge is contained in a unique 4-clique . this can not generalize to large values of @xmath63 , since by the removal lemma in a @xmath46-saturated graph on @xmath63 vertices with the property that every edge is contained in at most a constant number of 4-cliques , the number of edges is @xmath8 . the removal lemma argument that was used in @xcite to derive asymptotic lower bounds for the size of a maximal @xmath57-antichain in terms of turn numbers can be adapted to establish that for sets @xmath1 containing @xmath145 the construction in section [ subsec : general_construction ] gives the correct leading term . [ thm : strong_max_asymptotics ] let @xmath1 be a fixed set containing 2 and 3 , and let @xmath53 be its maximal element . for @xmath1-saturated graphs @xmath62 on @xmath63 vertices and @xmath186 , we have for @xmath193 , @xmath194    let @xmath185 be a pair of a @xmath1-saturated graph @xmath70 and an antichain @xmath186 which maximizes @xmath195 . in particular , @xmath196 , and since every triangle is contained in an element of some @xmath197 , and every @xmath14-clique contains @xmath198 triangles , the number of triangles in @xmath62 is bounded by @xmath199 by the removal lemma , we can make @xmath62 triangle - free by removing @xmath8 edges , in other words there is a partition @xmath200 such that @xmath201 and @xmath202 is triangle - free , and thus @xmath203 by turn s theorem . for @xmath204 and @xmath20 , let @xmath205 denote the number of elements of @xmath197 containing @xmath206 . using the @xmath1-saturation of @xmath62 and observing that @xmath202 , being triangle - free , contains at most @xmath207 edges of any @xmath14-clique in @xmath62 , we obtain @xmath208 and consequently @xmath209    [ cor : k_contains_3 ] let @xmath1 be a fixed set containing 2 and 3 with maximal element @xmath53 , and let @xmath17 be a maximal @xmath1-antichain of minimum size on @xmath63 points . then @xmath210    the lower bound follows from theorem [ thm : strong_max_asymptotics ] and the upper bound comes from the construction in section [ subsec : general_construction ] . [ cor : strongly_maximal_asymptotics ] let @xmath1 be a fixed set containing 2 with maximal element @xmath53 , and let @xmath17 be a strongly maximal @xmath1-antichain of minimum size on @xmath63 points . then @xmath210    the lower bound follows from corollary [ cor : k_contains_3 ] , because @xmath17 is also a maximal @xmath211-antichain , and the upper bound comes again from the construction in section [ subsec : general_construction ] . we expect that the statement of theorem [ thm : strong_max_asymptotics ] remains true if the condition @xmath7 is dropped , but in this case we can prove only a weaker bound . the main difficulty comes from the fact that we are not able to deduce that an optimal graph is essentially triangle - free ( meaning the number of triangles is @xmath212 ) . this weakens both bounds that are essential in the proof of theorem [ thm : strong_max_asymptotics ] : the upper bound for the number of edges , and the lower bound for the number of cliques in terms of the edge number . the obvious starting point for deriving bounds is that an optimal graph is almost @xmath213-free for every @xmath20 , which by turn s theorem yields an asymptotic upper bound for the number of edges . unfortunately , for @xmath9 this bound is far away from the conjectured truth @xmath214 . we now consider @xmath10 and derive a bound for @xmath215 using lower bounds for the number of triangles in graphs with more than @xmath214 edges . note that for sets @xmath1 of size @xmath153 , the antichain @xmath100 is the only element of @xmath82 , so maximizing @xmath215 is equivalent to ( [ eq : reformulation ] ) . for notational convenience , in the following we write @xmath216 for @xmath217 . proceeding as in the proof of theorem [ thm : strong_max_asymptotics ] , we obtain a first bound . [ lem : first_bound ] let @xmath62 be a @xmath46-saturated graph on @xmath63 vertices . then @xmath218    as @xmath219 , @xmath62 can be made @xmath220-free by removing @xmath8 edges , and we obtain a partition @xmath200 with @xmath221 and @xmath202 @xmath220-free , in particular any 4-clique in @xmath62 contains at most 5 edges from @xmath202 . using the 4-saturation of @xmath62 , we obtain @xmath222 and finally @xmath223 .    note that this proof together with turn s theorem already implies @xmath224 for @xmath46-saturated graphs . in order to improve the inequality , we bound the number of triangles . first we have a lower bound in terms of the number of edges . in any graph @xmath62 with @xmath63 vertices and @xmath225 edges , the number of triangles is at least ( see @xcite ) @xmath226 on the other hand we get an upper bound for the number of triangles in terms of the number of 4-cliques . [ lem : triples_and_quads ] let @xmath62 be a @xmath46-saturated graph on @xmath63 vertices . then the number of triangles is at most @xmath227    let @xmath228 denote the set of triangles that are contained in a 4-clique in @xmath62 , and let @xmath229 be the set of all other triangles . note that @xmath230 , so the number of triangles is dominated by @xmath231 . consider four vertices @xmath232 inducing a 4-clique in g , and let @xmath233 be another vertex . there is at most one triangle in @xmath229 of the form @xmath234 with @xmath235 . consequently , there are at most @xmath236 triangles in @xmath229 having two vertices in @xmath237 , and by summation over all 4-cliques the claim follows .    combining the two bounds for the number of triangles we obtain a second bound for @xmath215 in terms of @xmath216 . [ lem : second_bound ] let @xmath62 be a @xmath46-saturated graph on @xmath63 vertices . then @xmath238    now we can combine ( [ eq : first_bound ] ) and ( [ eq : second_bound ] ) to eliminate @xmath216 and obtain an absolute bound for @xmath215 . [ thm : weak_max_asymptotics ] let @xmath62 be a @xmath46-saturated graph on @xmath63 vertices . then @xmath239    the right hand sides of ( [ eq : first_bound ] ) and ( [ eq : second_bound ] ) are increasing and decreasing in @xmath216 , respectively . equating them and solving for @xmath216 yields @xmath240 and the result follows . let @xmath17 be a maximal @xmath46-antichain of minimum size on @xmath63 points . then @xmath241    the lower bound is a consequence of theorem [ thm : weak_max_asymptotics ] , and the upper bound comes from the construction in section [ subsec : l_4 ] . an obvious next step is to improve the asymptotic bounds with the aim to replace the coefficient of @xmath242 in theorem [ thm : weak_max_asymptotics ] by @xmath243 , which gives the leading term of the inequality in conjecture [ conj : graph_conjecture ] . it seems that arguments based on the removal lemma are not sufficient to reach this goal . one reason for this failure could be that the removal lemma only requires that there are only @xmath244 4-cliques , while in our situation we can use the stronger information that the numbers 4-cliques is @xmath245 . another problem is to characterize the extremal graphs ( resp . antichains ) up to isomorphism . for the case @xmath56 this was done in @xcite , and it would be nice to prove that for @xmath246 and large @xmath63 ( some slight variations of ) the constructions in section [ subsec : l_4 ] yield all extremal graphs . in view of the computational results in table [ tab : small_cases ] one might also study the question if any maximal @xmath1-antichain of minimum size contains only @xmath14-sets and @xmath53-sets , where @xmath14 and @xmath53 are the smallest and the largest element of @xmath1 , respectively . d.  gerbner , b.  keszegh , n.  lemons , c.  palmer , d.  plvlgyi , c.  palmer , and b.  patks . saturating sperner families . , pages 110 , online first : http://dx.doi.org/10.1007/s00373-012-1195-6[`doi:10.1007/s00373-012-1195-6 ` ] , 2012 . preprint http://arxiv.org/abs/1105.4453v1[`arxiv:1105.4453v1 ` ] . m.  grttmller , i.t . roberts , and l.j . rylands . antichains and completely separating systems - a catalogue and applications . , 2012 . in press , http://dx.doi.org/10.1016/j.dam.2012.04.016[`doi:10.1016/j.dam.2012.04.016 ` ] .
let @xmath0 be a natural number , and let @xmath1 be a set @xmath2:=\{1,2,\dots , n\}$ ] . 
 we study the problem to find the smallest possible size of a maximal family @xmath3 of subsets of @xmath4 $ ] such that @xmath3 contains only sets whose size is in @xmath1 , and @xmath5 for all @xmath6 , i.e. @xmath3 is an antichain . 
 we present a general construction of such antichains for sets @xmath1 containing 2 , but not 1 . if @xmath7 our construction asymptotically yields the smallest possible size of such a family , up to an @xmath8 error . 
 we conjecture our construction to be asymptotically optimal also for @xmath9 , and we prove a weaker bound for the case @xmath10 . 
 our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory which is interesting in its own right . 
 * keywords : * extremal set theory , sperner property , maximal antichains , flat antichains