On graphs with subgraphs of large independence numbers
aa r X i v : . [ m a t h . C O ] J un On graphs with subgraphs of large independence numbers
Noga Alon ∗ Benny Sudakov † Abstract
Let G be a graph on n vertices in which every induced subgraph on s = log n vertices hasan independent set of size at least t = log n . What is the largest q = q ( n ) so that every such G must contain an independent set of size at least q ? This is one of several related questions raisedby Erd˝os and Hajnal. We show that q ( n ) = Θ(log n/ log log n ), investigate the more generalproblem obtained by changing the parameters s and t , and discuss the connection to a relatedRamsey-type problem. What is the largest f = f ( n ) so that every graph G on n vertices in which every induced subgraphon log n vertices has an independent set of size at least log n , must contain an independent set ofsize at least f ? This is one of several related questions considered by Erd˝os and Hajnal in the late80s. The question appears in [3], where Erd˝os mentions that they thought that f ( n ) must be at least n / − ǫ , but they could not even prove that it is at least 2 log n . As a special case of our main resultshere we determine the asymptotic behavior of f ( n ) up to a factor of log log n , showing that in factit is much smaller than one may suspect (and yet much bigger than log n ):Ω( log n log log n ) ≤ f ( n ) ≤ O (log n ) . (1)Another specific variant of the above question, discussed in [3], is the problem of estimating thelargest q = q ( n ) so that every graph G on n vertices in which every induced subgraph on log n vertices has an independent set of size at least log n , must contain an independent set of size at least q . Here, too, one may tend to believe that q ( n ) is large, and specifically it is mentioned in [3] that ∗ Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, TelAviv University, Tel Aviv 69978, Israel. Email: [email protected]. Research supported in part by the Israel ScienceFoundation, by a USA-Israeli BSF grant and by the Hermann Minkowski Minerva Center for Geometry at Tel AvivUniversity. † Department of Mathematics, Princeton University, Princeton, NJ 08544, and Institute for Advanced Study, Prince-ton. E-mail: [email protected]. Research supported in part by NSF CAREER award DMS-0546523, NSFgrant DMS-0355497, USA-Israeli BSF grant, Alfred P. Sloan fellowship, and the State of New Jersey. q ( n ) > log n , but the correct asymptotic behavior of q ( n ) is smaller. In this case, ourresults determine the asymptotic behavior of q ( n ) up to a constant factor, implying that q ( n ) = Θ( log n log log n ) . (2)Both problems above are special instances of the general problem of understanding the asymptoticbehavior of the function f ( n, s, t ) defined as follows. For n > s > t , let f = f ( n, s, t ) denote thelargest integer f so that every graph G on n vertices in which every induced subgraph on s verticeshas an independent set of size at least t , must contain an independent set of size at least f . In thisnote we investigate the asymptotic behavior of f , and obtain rather tight bounds for this behaviorfor most interesting values of the parameters. Our results provide much less satisfactory informationabout a closely related Ramsey-type problem of Erd˝os and Hajnal discussed in [3], which is thefollowing. For which functions h ( n ) and g ( n ), where n > g ( n ) ≥ h ( n ) ≫
1, is there a graph on n vertices in which every induced subgraph on g ( n ) vertices contains a clique of size h ( n ) as well asan independent set of size h ( n ) ? In particular, Erd˝os and Hajnal conjectured that there is no suchgraph for g ( n ) = log n and h ( n ) = log n ; our results here do not settle this conjecture, and onlysuffice to show that there is no such graph with g ( n ) = c log n/ log log n and h ( n ) = log n , for someabsolute positive constant c .The rest of this note is organized as follows. In Section 2 we state our main results concerningthe behavior of the function f ( n, s, t ). The proofs are described in Section 3. The final Section 4contains a few remarks, including the (simple) connection between the study of f and the Ramsey-type question discussed above.Throughout the note we omit all floor and ceiling signs, whenever these are not crucial. Wealways assume that the number n of vertices of the graphs considered here is large. All logarithmsare in the natural base e . The following two theorems provide lower bounds for the independence numbers of graphs in whichevery induced subgraph of size s contains an independent set of size t . Theorem 2.1
Let t < s < n/ , and let G be a graph of order n such that every induced subgraphof G on s vertices contains an independent set of size t . Denote k = ⌊ st − ⌋ . Then G containsan independent set of size at least Ω (cid:0) kn /k (cid:1) if k ≤ n and of size at least Ω (cid:16) log n log( k/ log n ) (cid:17) if k > n . Theorem 2.2
Let t ≤ s < n/ , and let G be a graph of order n such that every induced subgraphof G on s vertices contains an independent set of size t . Then G contains an independent set of sizeat least Ω (cid:16) t log( n/s )log( s/t ) (cid:17) . s contains an independent set of size t . Theorem 2.3
For every sufficiently large t and t ≤ s ≤ n/ there exists a graph G on n verticeswith independence number α ( G ) ≤ O (cid:18) t (cid:16) ns (cid:17) t/ ( s − t ) log( n/t ) (cid:19) such that every induced subgraph of G of order s contains an independent set of size t . For certain values of t and s one can improve the previous result as follows. Theorem 2.4
Let t < s ≤ n/ , where s ≤ e t , and assume, further, that either there exists aconstant δ > such that ( s/t ) − δ ≥ log n or s/t = Ω(log n ) and there exists a constant γ > suchthat log t ≥ log γ n . Then there exists a graph G on n vertices with α ( G ) ≤ O (cid:18) t log( s/t ) log( n/t ) (cid:19) such that every induced subgraph of G of order s contains an independent set of size t . Proof of Theorem 2.1.
Suppose that G contains t − U has size atleast s . Then the size of the largest independent set in U is bounded by t −
1, since an independentset can intersect each of the t − G . Therefore if G ′ is a graph obtained from G by deleting the vertices of t − | V ( G ′ ) | > n − s ≥ n/
2. We also have that the largest clique in G ′ has size at most k = ⌊ st − ⌋ . Otherwise, by the above discussion, G will contain t − k + 1, whose union has at least ( k + 1)( t − ≥ s vertices. To finish the proofwe apply the classical bound of Erd˝os and Szekeres [4] (see also [5]) for the usual graph Ramseynumbers. This result asserts that the maximum possible number of vertices in a graph with neithera clique of size k + 1 nor an independent set of size ℓ + 1 is at most (cid:0) k + ℓk (cid:1) . By this estimate, G ′ contains an independent set of size ℓ , where (cid:0) k + ℓk (cid:1) ≥ n/
2. Thus (cid:16) e ( k + ℓ ) k (cid:17) k > n/ k ≤ n we get α ( G ) ≥ ℓ ≥ Ω (cid:16) kn /k (cid:17) . On the other hand, if k > n we use the estimate (cid:16) e ( k + ℓ ) ℓ (cid:17) ℓ ≥ (cid:0) k + ℓℓ (cid:1) = (cid:0) k + ℓk (cid:1) ≥ n/
2. This gives that ℓ ≥ Ω (cid:16) log n log( k/ log n ) (cid:17) and completes the proof of thetheorem. (cid:3) Proof of Theorem 2.2.
Let r be the largest integer such that n (cid:18) t e s (cid:19) r − ≥ s.
3t is easy to check that r = Ω (cid:16) log( n/s )log( s/t ) (cid:17) . To prove the theorem we construct a sequence of pairwisedisjoint independent sets X , . . . , X r together with a sequence of nested subsets V = V ( G ) ⊃ V ⊃ . . . ⊃ V r − such that the following holds. Each X i is a subset of V i − of size t/ X i ∩ V i = ∅ , thereare no edges between X i and V i and | V i | ≥ t e s | V i − | for all 1 ≤ i ≤ r −
1. Then the union of allsets X i forms an independent set in G of size rt/ (cid:16) t log( n/s )log( s/t ) (cid:17) .Assuming the sets X j , V j have already been constructed for all j < i , construct X i and V i asfollows. Let m be the size of V i − . The inductive hypothesis implies that m ≥ (cid:18) t e s (cid:19) − i | V | ≥ n (cid:18) t e s (cid:19) r − ≥ s. Since every subset of order s in V i − contains t independent vertices and every independent set ofsize t belongs to at most (cid:0) m − ss − t (cid:1) subsets of size s we have that V i − contains at least (cid:0) ms (cid:1) / (cid:0) m − ts − t (cid:1) independent sets of size t . Therefore, using that m ≥ s ≥ t and (cid:0) ab (cid:1) ≤ (cid:0) eab (cid:1) b , we conclude that thereexist a subset X i of V i − of size t/ (cid:0) ms (cid:1)(cid:0) m − ts − t (cid:1)(cid:0) mt/ (cid:1) = m !( m − t )! ( s − t )! s ! (cid:18) mt/ (cid:19) − ≥ (cid:16) ms (cid:17) t (cid:18) t em (cid:19) t/ = (cid:18) mt es (cid:19) t/ = e (cid:0) t e s m (cid:1) t/ ! t/ ≥ (cid:18) t e s mt/ (cid:19) independent sets of size t . This implies that V i − contains at least (cid:0) t e s mt/ (cid:1) subsets of size t/ X i forms an independent set of size t . Let V i be the union of all these subsets. Bydefinition, for every vertex of V i there is an independent set which contains it together with X i ,so there are no edges from X i to V i . Also, it is easy to see that the size of V i must be at least t e s m = t e s | V i − | . This completes the construction step and the proof of the theorem. (cid:3) Proof of Theorem 2.3.
We prove the theorem by considering an appropriate random graph. Asusual, let G n,p denote the probability space of all labeled graphs on n vertices, where every edgeappears randomly and independently with probability p = p ( n ). We say that the random graphpossesses a graph property P almost surely , or a.s. for brevity, if the probability that G n,p satisfies P tends to 1 as n tends to infinity. Clearly, it is enough to show that there is a value of the edgeprobability p such that G n,p satisfies the assertion of the theorem with positive probability.Let p = e t (cid:0) ns (cid:1) − t/ ( s − t ) . We claim that almost surely every subset of G = G n,p of size s spansat most s / (2 t ) − s/ s which violatesthis assertion is at most P ≤ (cid:18) ns (cid:19)(cid:18) s / s / (2 t ) − s/ (cid:19) p s / (2 t ) − s/ ≤ (cid:16) ens (cid:17) s (cid:18) e ss − t tp (cid:19) s / (2 t ) − s/ ≤ (cid:16) ens (cid:17) s (cid:18) e (cid:16) ns (cid:17) − t/ ( s − t ) (cid:19) s / (2 t ) − s/ ≤ − s/ = o (1) , o (1)-term tends to zero as s tends to infinity). This implies that with high probabilityevery subgraph of G on s vertices has average degree d ≤ s/t −
1. Therefore by Tur´an’s theorem itcontains an independent set of size at least sd +1 ≥ t . On the other hand, it is well known (see, e.g.,[2]), that almost surely the independence number of G n,p is bounded by α ( G n,p ) ≤ O (cid:0) p − log np (cid:1) ≤ O (cid:18) t (cid:16) ns (cid:17) t/ ( s − t ) log( n/t ) (cid:19) . This implies that a.s. G n,p satisfies the assertion of the theorem and completes the proof. (cid:3) For the proof of Theorem 2.4 we need the following lemma.
Lemma 3.1
Let G = G s,p be a random graph, assume sp → ∞ and fix ǫ > . Then the probabilitythat the independence number of G is at most ǫp log( sp ) is less than e − s ( sp ) − ǫ/ . Proof (of lemma).
Let k = ǫp log( sp ). To prove the lemma we use the standard greedy algorithmwhich constructs an independent set by examining the vertices of the graph in some fixed order andby adding a vertex to the current independent set whenever possible. The behavior of this algorithmfor random graphs can be analyzed rather accurately (see, e.g., [2]). At iteration i of our procedurewe use the greedy algorithm to find a maximal (with respect to inclusion) independent set I i inthe remaining vertices of G . If | I i | ≥ k we stop. Otherwise we delete the vertices of I i from G and continue. We stop when the number of remaining vertices drops below s/
2. Note that duringiteration number i we only expose edges incident to I i , therefore the remaining vertices still forma truly random graph. Given a set I the probability that a fixed vertex of G is adjacent to somevertex of I is 1 − (1 − p ) | I | . Therefore the probability that a fixed I is maximal, where | I | ≤ k , isat most (cid:0) − (1 − p ) k (cid:1) s/ . By definition, when the iteration fails, the random graph must containa maximal independent set of size less than k (and hence also a set of size exactly k so that everyremaining vertex is adjacent to at least one of its members). Thus the probability of this event is atmost (cid:0) sk (cid:1)(cid:0) − (1 − p ) k (cid:1) s/ . Moreover, the outcomes of different iterations depend on disjoint sets ofedges and therefore are independent. Finally note that if G has no independent set of size k , thenthe number of iterations is at least s/ (2 k ). This implies that the probability of such an event isbounded by P ≤ (cid:18)(cid:18) sk (cid:19)(cid:0) − (1 − p ) k (cid:1) s/ (cid:19) s/ (2 k ) ≤ (cid:16) esk (cid:17) s/ e − s k (1 − p ) k ≤ e − Ω (cid:0) s ( sp )1 − ǫ log sp (cid:1) + s log( sp ) ≤ e − s ( sp ) − ǫ/ . (cid:3) Proof of Theorem 2.4.
Suppose that ( s/t ) − δ ≥ log n for some fixed δ > G = G n,p with p = δ log( s/t )2 t . (Note that p < s ≤ e t .) As already mentionedabove a.s. the independence number of this graph is bounded by O (cid:0) p log( np ) (cid:1) = O (cid:0) t log( s/t ) log( n/t ) (cid:1) .Here we used that log( s/t ) < s/t < n/t and hence log (cid:0) nt log( s/t ) (cid:1) < n/t ). Also, by Lemma 3.15with ǫ = δ/ G contains an induced subgraph of order s with no independentset of size δ p log( sp ) ≥ t is at most (cid:18) ns (cid:19) e − s ( sp ) − δ/ ≤ n s e − s log δ/ n = o (1) . Here we used that log( s/t ) → ∞ and hence ( sp ) − δ/ ≥ ( s/t ) − δ/ ≥ log δ/ n . Therefore withhigh probability G satisfies the first assertion of the theorem.To prove the second part of the theorem suppose that s/t = Ω(log n ) and log t ≥ log γ n for someconstant γ >
0. By the previous paragraph we can also assume that s/t ≤ log n . Let G = G n,p with p = γ log( s/t )640 t . Again we have that a.s. α ( G ) ≤ O (cid:0) t log( s/t ) log( n/t ) (cid:1) . Since sp = Ω(log n log log n ), theprobability that there is a subset of size s in G which spans at least 2 s p edges is bounded by (cid:18) ns (cid:19)(cid:18) s / s p (cid:19) p s p ≤ n s (cid:18) e p (cid:19) s p p s p = n s (cid:16) e (cid:17) s p ≤ e s log n e − Ω( s log n log log n ) = o (1) . Also, since s/t ≤ sp ≤ t o (1) , we have that the probability that G contains a subset of order k = Θ( sp )which spans more than k − γ/ edges is at most (cid:18) nk (cid:19)(cid:18) k / k − γ/ (cid:19) p k − γ/ ≤ n k ( kp ) k − γ/ ≤ e k log n e − Ω( k ( s/t ) − γ/ log t ) ≤ e k log n e − Ω( k log γ/ n ) = o (1) . Let H be an induced subgraph of G of order s . Since the number of edges in H is a.s. atmost 2 s p it contains an induced subgraph H ′ of order s/ d = 8 sp .By the above discussion, we also have that the neighborhood of every vertex of H ′ spans at most d − γ/ edges and therefore through every vertex in H ′ there are at most t ≤ d − γ/ triangles. Nowwe can use the known estimate (Lemma 12.16 in [2], see also [1] for a more general result) on theindependence number of a graph containing a small number of triangles. It implies that α ( H ′ ) ≥ . | V ( H ′ ) | d (cid:18) log d −
12 log t (cid:19) ≥ . s/ d (cid:18) log d −
12 log d − γ/ (cid:19) = γs d log d ≥ γs sp log(8 sp ) ≥ t. This shows that G a.s. satisfies the second assertion of our theorem and completes the proof. (cid:3) Theorems 2.1 and 2.3 show that if s/t = O (1) and t > f ( n, s, t ) = n Θ(1) , whereas if s/t ≫ t = n o (1) then f ( n, s, t ) = n o (1) , and if s/t ≥ Ω(log n ) and t ≤ (log n ) O (1) then f ( n, s, t ) ≤ (log n ) O (1) . Theorems 2.2 and 2.4 determine the asymptotic behavior of the function f ( n, s, t ) up to a constantfactor for a wide range of the parameters. We do not specify here all this range, and only observe6hat in particular, for every fixed µ > f ( n, log µ n, log n ) = Θ( log n log log n ) . For µ = 1 this implies(2). The estimate (1) follows from Theorems 2.2 and 2.3.The connection between the Ramsey-type question described in Section 1 and the function f isthe following simple fact. Claim: If n/ > s > t and ( t − f ( n/ , s, t ) ≥ s, (3)then there is no graph on n vertices in which every induced subgraph on s vertices contains a cliqueof size t and an independent set of size t . Proof:
Assuming there is such a graph G , observe that by the definition of f it contains anindependent set I of size f = f ( n/ , s, t ). Omit this set, and observe that the induced graph on theremaining vertices, assuming there are at least n/ I ofsize f . Repeating this process t − n/ > s ), we get an induced subgraph of G on min { n/ , ( t − f } ≥ s vertices, whichis ( t − t − t . This is a contradiction, proving the assertion of the claim.In particular, for t = log n and s = c log n/ log log n , where c is a sufficiently small positiveabsolute constant, it is not difficult to check that the assumption in (3) holds, by Theorem 2.2.It will be interesting to close the gap between our upper and lower bounds for the function f ( n, s, t ). It will also be interesting to know more about the Ramsey-type question of Erd˝os andHajnal described in the introduction, and in particular, to decide if there exists a graph on n verticesin which every induced subgraph on, say, log n vertices, contains a clique of size at least log n andan independent set of size at least log n . References [1] N. Alon, M. Krivelevich and B. Sudakov, Coloring graphs with sparse neighborhoods,
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