aa r X i v : . [ m a t h . C O ] J a n A note on stability for maximal F -free graphs D´aniel Gerbner ∗ Abstract
Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal K r +1 -free graphswith (1 − r ) n − o ( n r +1 r ) edges contain a complete r -partite subgraph on n − o ( n )vertices. This was very recently extended to odd cycles in place of K by Wang, Wang,Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtainsome other stability results along the way. One of the most basic questions of graph theory is the following: given a graph F , howmany edges can an n -vertex graph G have if it is F -free, i.e. G does not contain F as asubgraph? This quantity is denoted by ex( n, F ). Tur´an’s theorem [18] states that among n -vertex K r +1 -free graphs, the most edges are in the complete r -partite graph with eachpartite set of order ⌊ n/k ⌋ or ⌈ n/k ⌉ . This graph is now called the Tur´an graph and we denoteit by T r ( n ). We denote the number of edges of T r ( n ) by t r ( n ).The Erd˝os-Stone-Simonovits theorem [8, 10] states that it r ≥ F has chromaticnumber r + 1, then ex( n, F ) = (1 + o (1)) t r ( n ). Erd˝os and Simonovits [] showed that if an n -vertex graph G is F -free and has almost t r ( n ) edges, then its structure is very similar tothe structure of the Tur´an graph. This phenomenon is called stability and there are severalnon-equivalent stability theorems concerning the same graphs, where the differences comefrom the precise form of “almost”, “structure” and “very similar” in the previous sentence.In particular, the Erd˝os-Simonovits stability theorem [6, 7, 17] says that if G is F -free on n vertices with t r ( n ) − o ( n ) edges, then we can obtain T r ( n ) by adding and deleting o ( n )edges.Tyomkin and Uzzel [19] initiated the study of new stability questions. We say that a graphis F -saturated if it is F -free, but adding any new edge would create a copy of F . We also saythat G is maximal with respect to the F -free property. When studying ex( n, F ), one mightassume without loss of generality that the n -vertex F -free graph G is F -saturated, but if we ∗ Alfr´ed R´enyi Institute of Mathematics, E-mail: [email protected].
Research supported by the Na-tional Research, Development and Innovation Office – NKFIH under the grants FK 132060, KKP-133819,KH130371 and SNN 129364. G , this is a useful assumption. Consider a K r +1 -saturated graphwith close to t r ( n ) edges. Does it contain a large complete r -partite subgraph? Popielarz,Sahasrabuddhe and Snyder [14] answered this question with the following theorem. Theorem 1.1 (Popielarz, Sahasrabuddhe and Snyder [14]) . Let r ≥ be an integer. Every K r +1 -saturated graph G on n vertices with t r ( n ) − o ( n r +1 r ) edges contains a complete r -partitesubgraph on (1 − o (1)) n vertices. Moreover, there are K r +1 -saturated graphs on n verticeswith t r ( n ) − Ω( n r +1 r ) edges that do not contain a complete r -partite subgraph on (1 − o (1)) n vertices. Wang, Wang, Yang and Yuan [20] considered the same problem for odd cycles in placeof cliques and showed the following.
Theorem 1.2.
Let k ≥ be an integer. Every C k +1 -saturated graph G on n vertices with t ( n ) − o ( n ) edges contains a complete bipartite subgraph on (1 − o (1)) n vertices. Moreover,there are C k +1 -saturated graphs on n vertices with t ( n ) − Ω( n ) edges that do not containa complete bipartite subgraph on (1 − o (1)) n vertices. Here we study the same problem for other graphs. Let us start with a bold conjecture.
Conjecture 1.3.
Let r ≥ be an integer and F be a graph with chromatic number r + 1 .Then every F -saturated graph G on n vertices with t r ( n ) − o ( n r +1 r ) edges contains a complete r -partite subgraph on (1 − o (1)) n vertices. Observe that in case of forbidden cliques or cycles, the complete multipartite subgraphmust be an induced subgraph. This is not the case in general, as we discuss in Section 2.If the above conjecture holds, the term o ( n r +1 r ) cannot be improved in general by Theorem1.1. Moreover, every 3-chromatic graph F contains an odd cycle, thus in case r = 2, theterm o ( n ) cannot be improved for any graph F by Theorem 1.2.If Conjecture 1.3 does not hold, weaker versions still should. Let us propose two suchversions. We say that a vertex or edge of a graph is color-critical , if deleting that vertex oredge results in a graph with smaller chromatic number. Conjecture 1.4.
Let r ≥ be an integer and F be a graph with chromatic number r + 1 and a color-critical edge. Then every F -saturated graph G on n vertices with t r ( n ) − o ( n r +1 r ) edges contains a complete r -partite subgraph G ′ on (1 − o (1)) n vertices. We remark that in this case G ′ has to be an induced subgraph. K r and r -chromaticgraphs with a color-critical edge often behave similarly in extremal questions, see e.g. [15]for several stability results. Another reason to assume that this conjecture might hold, andit might be easier to prove this than Conjecture 1.3 is the following. The proofs of Theorems1.1 and 1.2 both start with finding a not necessarily complete r -partite graph with manyvertices and edges, using only the F -free property. After that, both proofs continue withshowing that o ( n ) vertices are incident to all the missing edges between the two partite sets,thus removing those vertices finishes the proof.2he first step of this proof holds for every graph in this generality, but we need a verydelicate version that fully uses the property of having t r ( n ) − o ( n ( r +1) /r edges. Fortunately,the version due to Popielarz, Sahasrabudhe and Snyder (Lemma 2.3 in [14]) easily extendsto graphs with a color-critical edge. The version in [14] uses a result of Andr´asfai, Erd˝osand S´os [3] that determined the largest possible minimum degree in an n -vertex K r +1 -freegraph that is not r -partite, and a result of Brouwer [4] that determined the largest possibleminimum number of edges in such graphs.Erd˝os and Simonovits [9] extended the result of Andr´asfai, Erd˝os and S´os, while Si-monovits [16] extended the result of Brouwer asymptotically to any r -chromatic graph witha color-critical edge in place of K r . Using those results instead, the lemma below easilyfollows by the same proof as Lemma 2.3 in [14]. Lemma 1.5.
Let r ≥ and F be an ( r + 1) -chromatic graph with a critical edge. Thenthere is a constant d F , depending only on F , such that the following holds. If < α is smallenough, n is large enough, and G is an n -vertex F -free graph with | E ( G ) |≥ t r ( n ) − αn , thenthere is a subset T ⊂ V ( G ) with | T |≤ d F αn such that G − T is ( r − -partite. We omit the proof of this lemma. We will prove Conjecture 1.4 for some 3-chromaticgraphs with a color-critical edge, but we will use another lemma instead, making this paperself-contained.
Theorem 1.6.
Let F be a 3-chromatic graph with a color-critical edge such that every edgehas a vertex that is contained in a triangle. Then Conjecture 1.3 holds for F . Let us state a third conjecture, which is implied by the first and implies the second.
Conjecture 1.7.
Let r ≥ be an integer and F be a graph with chromatic number r + 1 anda color-critical vertex. Then every F -saturated graph G on n vertices with t r ( n ) − o ( n r +1 r ) edges contains a complete r -partite subgraph on (1 − o (1)) n vertices. A reason to assume that this conjecture might hold is that we are able prove it in thespecial case r = 2 and the color-critical vertex is connected to every other vertex of F . Theorem 1.8.
Let F be a 3-partite graph with a vertex w that is connected to every othervertex of F . Then Conjecture 1.3 holds for F . In our results, we are only interested in the order of magnitude and make no effort tooptimize or even precisely state constant factors. We also assume basically everywhere that n is large enough, which means that there is a constant n depending on the parametersintroduced earlier such that n ≥ n .The rest of this paper is organized as follows. In Section 2, we prove some necessarylemmas and some unnecessary lemmas: related results that we do not use later. In Section3, we present the proof of Theorems 1.8 and 1.6.3 Lemmas and other results
The main reason to assume that Theorem 1.1 can be extended as in Conjecture 1.3 is thatstability results often extend in a similar way. Let us show an example.
Theorem 2.1 (Nikiforov, Rousseau) . For r ≥ there is a constant d r , depending only on r , such that the following holds. For every < α ≤ d r , every K r -free n -vertex graph G with at least ( r − r − − α ) n edges contains an induced r -chromatic graph G ′ of order at least (1 − α / ) n and with minimum degree at least ( r − r − − α / ) n . We can extend the above theorem to any r -chromatic graph. Proposition 2.2.
For r ≥ there is a constant d ′ r , depending only on r , such that thefollowing holds. Let F be an r -chromatic graph and n be large enough. For every < α ′ ≤ d ′ r ,every F -free n -vertex graph G with at least ( r − r − − α ′ ) n edges contains an r -chromatic graph G ′ of order at least (1 − α ′ ) / ) n and with minimum degree at least ( r − r − − α ′ ) / ) n .Proof. By a result of Alon and Shikhelman [2], for any ε >
0, if n is large enough, then any n -vertex F -free graph contains at most εn | V ( H ) | copies of K r . By the removal lemma, forany δ > δ > n -vertex graph contains at most εn | V ( H ) copies of K r , then we can delete at most δn edges to obtain a K r -free graph. Let d ′ r be any numbersmaller than d r from Theorem 2.1, δ ≤ d r − α ′ be a constant, ε be as needed to apply theremoval lemma, and n be large enough so that we can use the result of Alon and Shikhelman.Then we can apply the removal lemma and delete δn edges to obtain a K r -free graph. Thenwe can apply Theorem 2.1 to this graph to find the desired G ′ . (cid:4) The simple proof of the above proposition works for many other stability results concern-ing K r +1 . Let us mention another example without going into details: Kor´andi, Roberts andScott [11] considered K r +1 -free graphs with at least t r ( n ) − δ r n edges, and determined thelargest number of edges one may need to remove from such a graph to obtain an r -partitegraph. If we consider an F -free graph where F has chromatic number r + 1, then we delete o ( n ) edges first to remove the copies of K r +1 as in the above, and then apply their theoremto obtain an upper bound on the number of edges we additionally need to remove. Thisbound will not be sharp for two reason: we started with more edges (by o ( n ), and we alsoremoved those edges), and their K r +1 -free construction showing the sharpness of their resultmay contain F . If F contains K r +1 , the second problem does not occur, and we obtain anasymptotically sharp result.The above proof method, i.e. the combination of the result of Alon and Shikhelmanand the removal lemma does not help with Conjecture 1.3, as we have to remove almostquadratic many edges when using the removal lemma. We can prove a stronger lemma fora much smaller class of graphs. Lemma 2.3.
Let F be a -chromatic graph with a color-critical vertex and n be large enough.Let | V ( F ) | n < α < | V ( F ) | . If G is an n -vertex F -free graph with | E ( G ) |≥ ex( n, F ) − αn ,then there is a bipartite subgraph H of G with at least (1 − | V ( F ) | α ) n vertices, at least n, F ) − kαn edges and minimum degree at least (cid:16) − | V ( F ) | (cid:17) n such that every vertexof H is adjacent in G to at most | V ( F ) | vertices in the same partite set of H .Proof. Observe that F is a subgraph of K ,k,k for some k . Simonovits [16] showed that forthe complete ( r + 1)-partite graph K = K ,k,...,k we have ex( n, K ) ≤ t r ( n ) + kn (in facthe obtained a more general result, that implies an exact result for ex( n, K ), and he alsodescribed the asymptotic structure). This implies that ex( n, F ) ≤ n + kn .We start by removing vertices of small degree, like the proofs of Theorem 1.1 in [14] andTheorem 1.2 in [20]. Let G = G and given G i on n i = n − i vertices, if every vertex of G i has degree at least ( − k ) n i , then we let G ′ = G i . If there is a vertex v with degree lessthan ( − k ) n i , then we let G i +1 be the graph obtained from G i by deleting v . Let n ′ bethe number of vertices of G ′ . We have | E ( G ) |≤ | E ( G ′ ) | + n − n ′ − X i =0 ( 12 − k )( n − i ) . (1)The right hand side of (1) is at most n − k ( n − n ′ ) / kn ′ , while the left hand isat least n − αn . This shows that n ′ ≥ (1 − kα ) n .Let us consider a partition of G ′ into two parts A and B with the most edges betweenparts. By the Erd˝os-Simonovits stability theorem, there are o ( n ) edges inside the parts. Wealso have that if a vertex v is connected to d vertices in its part, say A , then it is connectedto at least d vertices in the other part B . On the other hand, v is connected to at least( − k ) n ′ − d vertices of B . This implies that d ≤ ( − k ) n ′ . Thus we have that v (andevery other vertex) is connected to at least ( − k ) n ′ vertices in the other part.Let us assume that | A |≥ | B | . We call a vertex in A good if it is connected to at most k n ′ vertices in its part, and bad otherwise. Observe that a good vertex has at least ( − k ) n ′ ≥| B |− k n ′ neighbors in B . If u ∈ A is connected to k good vertices in A , then these k + 1vertices have at least k common neighbors in B , thus there is a K ,k,k in G , a contradiction.Indeed, u has at least ( − k ) n ′ neighbors in B , and each good vertex in A is connectedto all but at most k of these vertices. thus the k + 1 vertices chosen from A have at least( − k k ) n ′ ≥ k common neighbors in B . This shows that every bad vertex has at least k n ′ − k bad neighbors in its part. If there exists a bad vertex, then there are Θ( n ) badvertices. Each of them is connected to Θ( n ) vertices in the same part, thus there are Θ( n )edges inside the parts, a contradiction.Therefore, we can assume that every vertex of A is good, thus every vertex in A isconnected to at least ( − k ) n ′ vertices on the other side. This in particular shows that( − k ) n ′ ≤ | A | , | B |≤ ( + k ) n ′ . Now we call a vertex in A good if it is connected toat most k n ′ vertices in its part, and bad otherwise. Then a good vertex in B has at least( − k ) n ′ ≥ | A |− k n ′ neighbors in A . By the same reasoning as for bad vertices in A ,we obtain that the existence of one bad vertex in B would imply the existence of Θ( n ) badvertices in B and Θ( n ) edges inside the parts, a contradiction. Thus we can assume thatevery vertex is good. 5ssume now that a vertex v ∈ A is connected to at least k vertices in A . Then v and k of its neighbors in A are each connected to all but at most k n ′ vertices of B . Thus thenumber of vertices in B that are not connected to some of them is at most k ( k + 1) n ′ .Therefore, there are at least k other vertices in B , those are common neighbors of the k + 1vertices picked earlier, hence they form a copy of K ,k,k , a contradiction.This shows that there are at most ( k − n < αn edges inside A and B . Let us deleteall the edges inside the parts A and B from G ′ to obtain H . Clearly we have deleted atmost 12 kαn edges to get G ′ and at most αn edges to get H . The bounds on the numberof vertices and the minimum degree of H are obvious. (cid:4) Recall that the bipartite graph we found is not necessarily induced. However, for somegraphs we can strengthen the above result.
Corollary 2.4.
Let F be a 3-chromatic graph with a critical vertex such that F can alsobe obtained from a bipartite graph by adding a matching into one of the parts. Let n belarge enough and | V ( F ) | n < α < | V ( F ) | . If G is an n -vertex F -free graph with | E ( G ) |≥ ex( n, F ) − αn , then there is an induced bipartite subgraph H ′ of G with at least (1 − α ) n vertices, at least ex( n, F k ) − kαn edges and minimum degree at least ( − | V ( F ) | ) n .Proof. Let G ′ be the graph as in the proof of Lemma 2.3. We will show that inside thepartite sets A, B of G ′ there is no matching with | V ( F ) | edges. Indeed, the 2 | V ( F ) | verticesof that matching would have a common neighbor on the other partite set by the mini-mum degree condition, but this way we find a copy of F , a contradiction. This and thebound on the maximum degree implies that the subgraph of G inside A and inside B has(2 | V ( F ) |− | V ( F ) |− | V ( F ) |− | V ( F ) | independent edges. See [1] and [5] for moreprecise bounds on the number of edges.Therefore, we can remove the endpoints of those O (1) edges to obtain H ′ . It is easy tosee that H ′ has the desired number of vertices, edges and minimum degree, using that n islarge enough. (cid:4) We remark that if F cannot be obtained from a bipartite graph by adding a matching intoone of the parts, then a similar strengthening is impossible. Indeed, if we add a matching toone of the parts of the Tur´an graph, the resulting graph is F -free, and we need to removeabout n/ Proof of Theorem 1.8.
Let n be large enough and G be an n -vertex F -saturated graph with t ( n ) − o ( n / ) edges. First we apply Lemma 2.3 with α = o ( n − / ) to obtain H with partitesets A and B . We will also use the subgraph G ′ of G from the proof of Lemma 2.3, that is H with additional edges inside the parts, such that every vertex is incident to at most | V ( F ) | such edges. Let T denote the set of vertices not in G ′ , thus | T | = o ( n / ). For v ∈ T , let A ( v )6enote its neighborhood in A and B ( v ) denote its neighborhood in B . Let U ( v ) denote thesmaller of A ( v ) and B ( v ) (if they have the same number of vertices, we choose one of themarbitrarily). Let U = ∪ v ∈ T U ( v ) and for 1 ≤ i ≤ | V ( F ) | , U i denotes the set of vertices thatare connected to a vertex of U i − in the same parts. As the degrees inside A and B are atmost | V ( F ) | , we have that | U | V ( F ) | |≤ | V ( F ) | | V ( F ) | | U | .Let us now consider a partition of F to two connected subgraphs F and F such that w (the vertex that is connected to every other vertex of F ) is in F . Let Q denote the subgraphinduced on the vertices in F that are connected to some vertices of F . Then w is in Q ,thus Q is also connected.Consider the copies of F inside A and those copies of Q inside B that can be extended toa copy of F in G (note that we do not care where the additional vertices come from or howmany such extensions exist). Observe that every vertex v ∈ A is contained in O (1) copiesof F . Indeed, as F is connected, there is a path of length at most | V ( F ) | from v to everyvertex of such copies, and there are at most | V ( F ) | | V ( F ) | vertices in A that can be reachedfrom v by a path of length at most | V ( F ) | that is totally inside A . If v ∈ B , then there are O (1) copies of Q containing it by the same reasoning (in fact there is a path of length atmost 2 from v to other vertices of Q in this case).Let us assume that there are at least n / copies of F inside A and at least n / copiesof Q inside B . For any such copy of Q , we pick an extension to F , and observe that itintersects O (1) copies of F inside A . For the other copies of F , there is a vertex u in thecopy of F and a vertex v in the copy of Q such that uv is not an edge in G , by the F -freeproperty. This way for n / − O ( n / ) pairs of F and F , we found a missing edge. As both u and v are counted O (1) times, this means that Ω( n / ) edges between A and B are missingfrom G . There are at most n / − Ω( n / ) edges of G between A and B , O ( n ) edges inside A and B , and at most n | T | = o ( n / ) edges of G are incident to T . Therefore, the total numberof edges of G is at most the sum of these, contradicting our assumption.We obtained that there are less than n / copies of either F in A or Q in B . We take thevertices of each to form the set U ′ . Then we repeat this with copies of F in B and copiesof Q in A to obtain U ′ , and then with every other bipartition of F into two connected partsto obtain sets U ′ i of vertices. Let U be the union of all the sets U j and U ′ i . Claim 3.1. | U | = o ( n ) .Proof. For every i , U ′ i has cardinality at most n / , and we have constant many of them,thus their total cardinality is O ( n / ). We also have O (1) copies of U j , each of cardinality O ( | U | ), thus it is enough to show that | U | = o ( n ).Let F ′ be the bipartite graph we obtain by deleting w from F . By the K˝ov´ari-T. S´os-Tur´an theorem [12], ex( n, F ′ ) = o ( n ), thus there exists an m such that if n ≥ m , thenex( n, F ′ ) ≤ n /
5. Consider a vertex v ∈ T . For the vertices v with | U ( v ) | < m , altogether atmost m | T | = o ( n ) vertices are in the sets U ( v ). Let T ′ denote the set of vertices v ∈ T with | U ( v ) |≥ m . We take | U ( v ) | vertices from both A ( v ) and B ( v ), and consider the bipartitegraph G ( v ) defined by the edges of H between these subsets. Clearly G ( v ) is F ′ -free, thusthere are at most 4 | U ( v ) | / G ′ between these two parts and at least | U ( v ) | / | U ( v ) | ) edges are missing between A ( v ) and B ( v ).7e know that in total o ( n / ) edges are missing between A and B , thus P v ∈ T ′ | U ( v ) | = o ( n / ). On the other hand, by the Cauchy-Schwartz inequality we have that P v ∈ T ′ | U ( v ) | ≥ ( P v ∈ T ′ | U ( v ) | ) / | T ′ | . This implies P v ∈ T ′ | U ( v ) | = o ( n ), finishing the proof. (cid:4) Let us return to the proof of the theorem. Let G be the graph we obtain by deleting thevertices of U from H . We will show that G is a complete bipartite graph with partite sets A ′ = A \ U and B ′ = B \ U . Assume indirectly that u ∈ A ′ , v ∈ B ′ and uv is not an edgeof G , thus not an edge of G . Then adding the edge uv to G creates a copy of F , which wedenote by F ∗ . The vertex w of F ∗ is either u , v , or connected to both u and v , thus cannotbe in T . Assume without loss of generality that w ∈ A .Let R denote the set of the neighbors of v in F ∗ , then they are either in A , or in B .But in B , elements of R cannot belong to U i with 0 ≤ i ≤ | V ( F ) |− v would bein U i +1 and not in B ′ ). Let R j for j < | V ( F ) | denote the set of neighbors of the vertices of R j − ∩ B in F ∗ . Then similarly, we have that elements of R j belong to A or B , and those in B cannot belong to U i with 0 ≤ i ≤ | V ( F ) |− j . R j stops increasing before we arrive to U ,let R denote the final R j obtained this way. Then R ∩ B induces a connected subgraph of F ∗ by construction. Let F denote this subgraph, F denote the remaining part of F ∗ and Q denote the subgraph of F ∗ induced on R ∩ A . Then this is a partition as described in theconstruction of U , thus we have moved each vertices of R ∩ A or R ∩ B to U . In particular u or v is in U and not in G , a contradiction. (cid:4) Proof of Theorem 1.6.
We start the proof similarly to that of Theorem 1.8, but the set U of vertices we delete will be slightly different. We apply Corollary 2.4 with α = o ( n − / )to obtain H ′ with partite sets A and B . Let T denote the set of vertices not in H ′ , thus | T | = o ( n / ). For each vertex v ∈ T , let A ( v ) denote its neighborhood in A , B ( v ) denote itsneighborhood in B and U ( v ) denote the smaller of A ( v ) and B ( v ) (an arbitrary one of themin case they have the same size), as in the proof of Theorem 1.8. We let U = ∪ v ∈ T U ( v ),thus | U | = o ( n ) as in the proof of Theorem 1.8.Observe that every vertex u ∈ A ( v ) is connected to less than | V ( F ) | vertices in B ( v ).Indeed, otherwise these | V ( F ) | vertices have | V ( F ) | other common neighbors in A by the min-imum degree condition, and these 2 | V ( F ) | vertices together with u and v form K | V ( F ) | , | V ( F ) | +2 with an additional edge in one of the parts. This subgraph clearly contains F , a contradic-tion.For every v ∈ T , this means that the vertices of U ( v ) have at most | V ( F ) || U ( v ) | commonneighbors with v in the other partite set of H ′ . Let U ′ ( v ) be the set of those commonneighbors and let U ′ = ∪ v ∈ T U ′ ( v ). Then | U ′ |≤ | V ( F ) || U | = o ( n ).Consider now an edge uv inside T . We have that u and v have less than | V ( F ) | commonneighbors in A (and at most | V ( F ) | common neighbors in B ) by the same reasoning: oth-erwise we can find | V ( F ) | common neighbors of those vertices in B by the minimum degreecondition, giving us a copy of K | V ( F ) | , | V ( F ) | +2 , a contradiction. Let U ′′ ( uv ) denote the setof these less than 2 | V ( F ) | common neighbors and let U ′′ = ∪ u,v ∈ T,uv ∈ E ( G ) U ′′ ( uv ). Clearly | U ′′ | = o ( n ).Let U = U ∪ U ′ ∪ U ′′ . We delete U from H ′ to obtain G .8ssume that u ∈ A \ U ′ , v ∈ B \ U ′ and uv is not an edge in G , thus not an edge in G . Then adding the edge uv to G creates a copy of F that we denote by F ∗ . There is atriangle containing u or v , say uxy in F ∗ . One of its vertices, say x is in T , since H ′ isbipartite. Then U ( x ) = B ( x ), since u U . As y is connected to u , we have y ∈ B ∪ T . If y ∈ B , then y ∈ U , but then its common neighbors with x , including u , were moved to U ′ ,a contradiction. Thus y ∈ T , but then u ∈ U ′′ ( xy ) ⊂ U , a contradiction. (cid:4) References [1] H. L. Abbott, F. Hanson, N. Sauer. Intersection theorems for systems of sets.
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