A construction of Alon yields a sequence of highly pseudorandom triangle-free
graphs with edge density significantly higher than one might expect from
comparison with random graphs. We give an alternative construction for such
graphs.Comment: 6 page

Conlon, David

English

arXiv

A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs

Caltech Authors - Main

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A sequence of triangle-free pseudorandom graphs
David Conlon∗
Abstract
A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge
density significantly higher than one might expect from comparison with random graphs. We give
an alternative construction for such graphs.
1 Introduction
A graph G is said to be (p, β)-jumbled if
∣∣∣∣e(X)− p
(|X|
2
)∣∣∣∣ ≤ β|X|
for all X ⊆ V (G). For example, the binomial random graph Gn,p is (p, β)-jumbled with β = O(√pn).
It is not hard to show [5, 6] that this is essentially best possible, in that a graph with n vertices
cannot be (p, β)-jumbled with β = c
√
pn for c sufficiently small. For further information on jumbled
graphs and their properties, we refer the reader to the survey [8] or, for more recent developments,
the paper [4].
One important class of (p, β)-jumbled graphs is the collection of (n, d, λ)-graphs. These are d-regular
graphs on n vertices such that all eigenvalues of the adjacency matrix, save the largest, are bounded
in absolute value by λ. By the famous expander mixing lemma, these graphs are (p, β)-jumbled with
p = d/n and β = λ.
One of the best known examples of a pseudorandom graph, constructed by Alon [1], is a triangle-
free (n, d, λ)-graph with n = 23k, d = 2k−1(2k−1 − 1) and λ = O(2k). Taking p = d/n, we have√
pn =
√
d = Ω(2k), so the graph is close to optimally pseudorandom. Since p = Ω(n−1/3), the
construction also has surprisingly high density. While there are various ways to modify the usual
random graph to produce triangle-free graphs with density roughly n−1/2 (see, for example, [3]), no
such modification can hope to push very far past this density. Nevertheless, Alon’s construction does
so. The purpose of this note is to give another construction for such graphs.
Theorem 1 There exists a sequence (ni)
∞
i=1 of positive integers such that, for each i ≥ 1, there is a
triangle-free graph Gi on ni vertices which is (p, β)-jumbled with p = Ω(n
−1/3
i ) and β = O(
√
pni log ni).
Our construction is weaker than Alon’s on several counts: it does not produce regular graphs; it is
not completely explicit; and it does not generalise easily. One might also level the accusation that
the resulting graphs are not optimally pseudorandom, with the condition β = O(
√
pn log n) being a
∗Mathematical Institute, Oxford OX2 6GG, United Kingdom. Email: david.conlon@maths.ox.ac.uk. Research
supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632.
1
logarithmic factor away from the desired bound. However, it seems likely that this extra log factor
is simply an artifact of our proof. Countering these disadvantages, we believe that our construction,
which we describe in detail below, is more intuitive than Alon’s.
For concreteness, we will work with the polarity graph of Lazebnik, Ustimenko and Woldar [9], though
the role played by this graph could also be taken by a number of other C6-free graphs with n vertices
and Ω(n4/3) edges. Suppose then that q is an odd power of 2 and let n = q3+ q2+ q+1. The polarity
graph, which has a small number of loops, is an (n, d, λ)-graph with d = q + 1 ≥ n1/3 and, as noted
in [8, Section 3.7], λ =
√
2q = O(n1/6). Once the loops are removed, the resulting graph, which we
label by H, is C3, C4 and C6 free. For each vertex v in H, we randomly partition its neighbourhood
NH(v) into two sets Av and Bv and let Gv be the complete bipartite graph between Av and Bv. We
now define G to be the graph with the same vertex set as H and edge set ∪v∈V (H)Gv. We will show
that G asymptotically almost surely satisfies the requirements of Theorem 1.
It is straightforward to verify that G contains no triangles. To begin, note that since H is C4-free, the
only edges of G in NH(v) are those in Gv. Now suppose that abc is a triangle in G. If a, b and c are
all contained in the same neighbourhood NH(v) they cannot form a triangle, since the only edges of
G in NH(v) are those in Gv and Gv is bipartite. It must then be the case that the edges ab, bc and ca
are contained in three different neighbourhoods NH(u), NH(v) and NH(w), respectively. If u, v and
w are all distinct from a, b and c, then aubvcw would form a cycle of length 6 in H, contradicting the
fact that H is C6-free. On the other hand, if u = c, say, then H must contain the triangle bvc, again
contradicting the choice of H.
The remaining claim, that G is asymptotically almost surely (p, β)-jumbled with p = Ω(n−1/3) and
β = O(
√
pn log n), will be verified in the next section.
2 Proving G is jumbled
Let H0 be the polarity graph with n = q
3+ q2+ q+1 vertices. This graph is (q+1)-regular, has q2+1
loops and all eigenvalues of the adjacency matrix, save the largest, are bounded in absolute value by√
2q. We will form G from H0 by a slightly different procedure to that described in the introduction,
though the two are easily seen to produce the same graph.
In the first step, we form H
(2)
0 , the multigraph with loops on the same vertex set as H0 where two
vertices are joined if there is a walk of length two in H0 between them, allowing for multiple edges if
there is more than one such walk. We note that each vertex has q + 1 loops, one for each edge in H0,
and, since H, the simple graph formed by removing the loops from H0, is C3 and C4-free, the only
parallel edges arise from loops in H0. In the next step, we turn H
(2)
0 into a simple graph by removing
all loops from H
(2)
0 and all edges whose corresponding walk in H0 used a loop. The resulting graph
G1 is easily seen to be the union of n cliques, each clique being NH(v) for some v in H. We now form
the required graph, as before, by randomly partitioning NH(v) into two sets Av and Bv and letting G
be the union over all v in H of the complete bipartite graphs between Av and Bv.
Following this plan, we first look at H
(2)
0 . Letting M be the adjacency matrix of H0, the adjacency
matrix of H
(2)
0 is simply M
2, which implies that the eigenvalues of H
(2)
0 are the squares of the eigen-
values of H0. Therefore, H
(2)
0 is an (n, d, λ)-graph with d = (q + 1)
2 and λ = 2q. Since each vertex of
H
(2)
0 is contained in exactly q + 1 loops, the graph G0 formed by removing these loops has adjacency
matrixM2−(q+1)I, implying that G0 is an (n, d, λ)-graph with d = q(q+1) and λ = q+1. Therefore,
2
by the expander mixing lemma, for all X ⊆ V (G0),∣∣∣∣eG0(X)− qq2 + 1
(|X|
2
)∣∣∣∣ ≤ (q + 1)|X|,
where we used the fact that n = q3 + q2 + q + 1 = (q2 + 1)(q + 1).
Note now that every loop inH0 has exactly q other neighbours with which it can form a non-degenerate
walk of length two. Therefore, since any X ⊆ V (H0) contains at most |X| loops, we remove at most
q|X| edges from X when forming G1 from G0. By the estimate above, this implies that, for all
X ⊆ V (G1), ∣∣∣∣eG1(X)− qq2 + 1
(|X|
2
)∣∣∣∣ ≤ (2q + 1)|X|.
Therefore, G1 is (p, β)-jumbled with p = q/(q
2 + 1) and β = 2q + 1.
Recall that G1 is the union of cliques, while G is the union of random bipartite graphs, one for each
clique in G1. Our aim now is to show that asymptotically almost surely G is (p, β)-jumbled with
p = q/2(q2 + 1) = Ω(n−1/3) and β = O(q log n) = O(
√
pn log n). We will do this in a rather naive
fashion, estimating the probability that
∣∣∣∣eG(X) − q2(q2 + 1)
(|X|
2
)∣∣∣∣ ≤ Cq|X| log n (1)
for any given X and taking a union bound. To do this, we will need the following concentration
inequality for quadratic forms in independent random variables due to Hanson and Wright [7] (the
exact version we state follows from Theorem 1.1 in [10]).
Lemma 1 Let Z = (Z1, . . . , Zt) ∈ {−1,+1}t be a random vector with independent components each
of which is equal to 1 or −1 with probability 1/2. Let M be a t× t real matrix. Then
P[|ZTMZ − E(ZTMZ)| > ǫ] ≤ 2 exp
{
−cmin
(
ǫ2
‖M‖2F
,
ǫ
‖M‖
)}
,
where ‖M‖F = (
∑
i,jm
2
ij)
1/2 is the Frobenius norm and ‖M‖ = supx 6=0 ‖Mx‖2/‖x‖2 is the spectral
norm.
Suppose now that G[X] is the union of s cliques T1, . . . , Ts, of orders t1, . . . , ts, and let t = t1+ · · ·+ ts.
We define t random variables X1, . . . ,Xt, each equal to 1 or −1 with probability 1/2, and assign one
of these random variables to every vertex of every clique, noting that any given vertex may receive
multiple random variables, but only one relative to any given clique.
Suppose that the random variables assigned to the clique Ti are Zi1, . . . , Ziti . If vi is the vertex whose
neighbourhood in H is Ti, the value of Zij determines whether its corresponding vertex Ti(j) is placed
in Avi or Bvi , with Ti(j) placed in Avi if Zij = 1 and Bvi if Zij = −1. The number of edges in G[Ti]
is then
eG(Ti) = |Avi ||Bvi | =

ti
2
+
1
2
ti∑
j=1
Zij



 ti
2
− 1
2
ti∑
j=1
Zij

 = t2i
4
− 1
4
ti∑
j=1
ti∑
k=1
ZijZik.
3
Summing over i, we have
eG(X) =
s∑
i=1
eG(Ti) =
s∑
i=1
t2i
4
− 1
4
s∑
i=1
ti∑
j=1
ti∑
k=1
ZijZik
=
1
2
s∑
i=1
(
ti
2
)
+
1
4
s∑
i=1
ti − 1
4
s∑
i=1
ti∑
j=1
ti∑
k=1
ZijZik
=
eG1(X)
2
− 1
4
s∑
i=1
∑
1≤j 6=k≤ti
ZijZik.
Therefore, by our estimate on eG1(X), it only remains to show that
Q =
s∑
i=1
∑
1≤j 6=k≤ti
ZijZik
is smaller than Cq|X| log n with sufficiently high probability.
Let M be the t× t matrix whose entry mjk is equal to 1 if j 6= k and j and k (each of which represents
a vertex associated to a particular clique) are from the same clique Ti. Otherwise, we take mjk to be
0. Then Q = ZTMZ and it follows from the Hanson–Wright bound that
P[|Q| > Cq|X| log n] ≤ 2 exp
{
−cmin
(
C2q2|X|2 log2 n
‖M‖2F
,
Cq|X| log n
‖M‖
)}
.
But it is straightforward to verify that
‖M‖2F =
s∑
i=1
ti(ti − 1) = 2eG1(X) ≤
|X|2
q
+ 6q|X| ≤ 2max
{ |X|2
q
, 6q|X|
}
and, writing ρ(A) for the spectral radius of a matrix A,
‖M‖ =
√
ρ(M∗M) = ρ(M) ≤ sup
x 6=0
‖Mx‖∞
‖x‖∞ = max1≤j≤t
t∑
k=1
|mjk| ≤ q.
Therefore,
P[|Q| > Cq|X| log n] ≤ 2 exp
{
−cmin
(
1
2
C2q3 log2 n,
1
12
C2q|X| log2 n,C|X| log n
)}
≤ 2 exp{−2|X| log n}
for C sufficiently large in terms of c. Applying the union bound, we see that the probability there
exists a set X such that (1) fails is at most
n∑
|X|=1
(
n
|X|
)
2e−2|X| logn ≤ 2
n∑
|X|=1
n|X|e−2|X| logn ≤ 2
n∑
|X|=1
e−|X| logn ≤ 2
n− 1 .
The result follows.
4
3 Concluding remarks
Our construction also extends to give a sequence (ni)
∞
i=1 of positive integers such that, for each
i ≥ 1, there is a C5-free graph Gi on ni vertices which is (p, β)-jumbled with p = Ω(n−3/5i ) and
β = O(
√
pni log ni). The construction starts with the C10-free polarity graph of Lazebnik, Ustimenko
and Woldar [9], but follows the proof of Theorem 1 in all other respects. Because we lack optimal
constructions of C2ℓ-free graphs for ℓ ≥ 6, our method does not extend further to give constructions
of pseudorandom C2k+1-free graphs for k ≥ 3. As mentioned in the introduction, this is a distinct
weakness of our method when compared to Alon’s, which does extend to longer odd cycles [2, 8].
An alternative method for proving the jumbledness of our construction G might be to estimate the
eigenvalues of its adjacency matrix M by using the fact that Tr(M2k) is both the sum of the 2kth
powers of its eigenvalues and the number of walks of length 2k in G. However, G is constructed
by starting from a graph G1 which is a union of cliques and then taking a random bipartite graph
within each clique. This process causes an imbalance between odd and even cycles within each clique,
deleting all odd cycles, while doubling the proportion of even cycles relative to the density. This makes
it difficult to count the number of degenerate walks of length 2k without having close control over the
counts of degenerate walks of different types in the base graph G1. Nevertheless, it is plausible that
this could be done, and may even allow one to save the lost logarithmic factor in β.
Another definite weakness of our method is that it is not explicit and so, unlike Alon’s example, cannot
be used to give a constructive lower bound for the off-diagonal Ramsey number r(3, t). It remains
to decide whether there is some more explicit method for choosing large bipartite subgraphs of the
cliques in G1 which also produces a highly pseudorandom subgraph.
Acknowledgements. I would like to thank the anonymous referee for a number of helpful remarks.
References
[1] N. Alon, Explicit Ramsey graphs and orthonormal labelings, Electron. J. Combin. 1 (1994), R12,
8pp.
[2] N. Alon and N. Kahale, Approximating the independence number via the θ-function, Math. Pro-
gram. 80 (1998), 253–264.
[3] T. Bohman, The triangle-free process, Adv. Math. 221 (2009), 1653–1677.
[4] D. Conlon, J. Fox and Y. Zhao, Extremal results in sparse pseudorandom graphs, Adv. Math. 256
(2014), 206–290.
[5] P. Erdo˝s, M. Goldberg, J. Pach and J. Spencer, Cutting a graph into two dissimilar halves, J.
Graph Theory 12 (1988), 121–131.
[6] P. Erdo˝s and J. Spencer, Imbalances in k-colorations, Networks 1 (1971/72), 379–385.
[7] D. L. Hanson and F. T. Wright, A bound on tail probabilities for quadratic forms in independent
random variables, Ann. Math. Stat. 42 (1971), 1079–1083.
[8] M. Krivelevich and B. Sudakov, Pseudo-random graphs, inMore Sets, Graphs and Numbers, Bolyai
Soc. Math. Stud., Vol. 15, 199–262, Springer, Berlin, 2006.
5
[9] F. Lazebnik, V. A. Ustimenko and A. J. Woldar, Polarities and 2k-cycle-free graphs, Discrete
Math. 197/198 (1999), 503–513.
[10] M. Rudelson and R. Vershynin, Hanson–Wright inequality and sub-Gaussian concentration, Elec-
tron. Commun. Probab. 18 (2013), 1–9.
6


A Sequence of Triangle-Free Pseudorandom Graphs

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 A
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01
6
A sequence of triangle-free pseudorandom graphs
David Conlon∗
Abstract
A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge
density significantly higher than one might expect from comparison with random graphs. We give
an alternative construction for such graphs.
1 Introduction
A graph G is said to be (p, β)-jumbled if
∣
∣
∣
∣
e(X)− p
(|X|
2
)∣
∣
∣
∣
≤ β|X|
for all X ⊆ V (G). For example, the binomial random graph Gn,p is (p, β)-jumbled with β = O(
√
pn).
It is not hard to show [5, 6] that this is essentially best possible, in that a graph with n vertices
cannot be (p, β)-jumbled with β = c
√
pn for c sufficiently small. For further information on jumbled
graphs and their properties, we refer the reader to the survey [8] or, for more recent developments,
the paper [4].
One important class of (p, β)-jumbled graphs is the collection of (n, d, λ)-graphs. These are d-regular
graphs on n vertices such that all eigenvalues of the adjacency matrix, save the largest, are bounded
in absolute value by λ. By the famous expander mixing lemma, these graphs are (p, β)-jumbled with
p = d/n and β = λ.
One of the best known examples of a pseudorandom graph, constructed by Alon [1], is a triangle-
free (n, d, λ)-graph with n = 23k, d = 2k−1(2k−1 − 1) and λ = O(2k). Taking p = d/n, we have√
pn =
√
d = Ω(2k), so the graph is close to optimally pseudorandom. Since p = Ω(n−1/3), the
construction also has surprisingly high density. While there are various ways to modify the usual
random graph to produce triangle-free graphs with density roughly n−1/2 (see, for example, [3]), no
such modification can hope to push very far past this density. Nevertheless, Alon’s construction does
so. The purpose of this note is to give another construction for such graphs.
Theorem 1 There exists a sequence (ni)
∞
i=1 of positive integers such that, for each i ≥ 1, there is a
triangle-free graph Gi on ni vertices which is (p, β)-jumbled with p = Ω(n
−1/3
i ) and β = O(
√
pni log ni).
Our construction is weaker than Alon’s on several counts: it does not produce regular graphs; it is
not completely explicit; and it does not generalise easily. One might also level the accusation that
the resulting graphs are not optimally pseudorandom, with the condition β = O(
√
pn log n) being a
∗Mathematical Institute, Oxford OX2 6GG, United Kingdom. Email: david.conlon@maths.ox.ac.uk. Research
supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632.
1
logarithmic factor away from the desired bound. However, it seems likely that this extra log factor
is simply an artifact of our proof. Countering these disadvantages, we believe that our construction,
which we describe in detail below, is more intuitive than Alon’s.
For concreteness, we will work with the polarity graph of Lazebnik, Ustimenko and Woldar [9], though
the role played by this graph could also be taken by a number of other C6-free graphs with n vertices
and Ω(n4/3) edges. Suppose then that q is an odd power of 2 and let n = q3+ q2+ q+1. The polarity
graph, which has a small number of loops, is an (n, d, λ)-graph with d = q + 1 ≥ n1/3 and, as noted
in [8, Section 3.7], λ =
√
2q = O(n1/6). Once the loops are removed, the resulting graph, which we
label by H, is C3, C4 and C6 free. For each vertex v in H, we randomly partition its neighbourhood
NH(v) into two sets Av and Bv and let Gv be the complete bipartite graph between Av and Bv. We
now define G to be the graph with the same vertex set as H and edge set ∪v∈V (H)Gv. We will show
that G asymptotically almost surely satisfies the requirements of Theorem 1.
It is straightforward to verify that G contains no triangles. To begin, note that since H is C4-free, the
only edges of G in NH(v) are those in Gv. Now suppose that abc is a triangle in G. If a, b and c are
all contained in the same neighbourhood NH(v) they cannot form a triangle, since the only edges of
G in NH(v) are those in Gv and Gv is bipartite. It must then be the case that the edges ab, bc and ca
are contained in three different neighbourhoods NH(u), NH(v) and NH(w), respectively. If u, v and
w are all distinct from a, b and c, then aubvcw would form a cycle of length 6 in H, contradicting the
fact that H is C6-free. On the other hand, if u = c, say, then H must contain the triangle bvc, again
contradicting the choice of H.
The remaining claim, that G is asymptotically almost surely (p, β)-jumbled with p = Ω(n−1/3) and
β = O(
√
pn log n), will be verified in the next section.
2 Proving G is jumbled
Let H0 be the polarity graph with n = q
3+ q2+ q+1 vertices. This graph is (q+1)-regular, has q2+1
loops and all eigenvalues of the adjacency matrix, save the largest, are bounded in absolute value by√
2q. We will form G from H0 by a slightly different procedure to that described in the introduction,
though the two are easily seen to produce the same graph.
In the first step, we form H
(2)
0 , the multigraph with loops on the same vertex set as H0 where two
vertices are joined if there is a walk of length two in H0 between them, allowing for multiple edges if
there is more than one such walk. We note that each vertex has q + 1 loops, one for each edge in H0,
and, since H, the simple graph formed by removing the loops from H0, is C3 and C4-free, the only
parallel edges arise from loops in H0. In the next step, we turn H
(2)
0 into a simple graph by removing
all loops from H
(2)
0 and all edges whose corresponding walk in H0 used a loop. The resulting graph
G1 is easily seen to be the union of n cliques, each clique being NH(v) for some v in H. We now form
the required graph, as before, by randomly partitioning NH(v) into two sets Av and Bv and letting G
be the union over all v in H of the complete bipartite graphs between Av and Bv.
Following this plan, we first look at H
(2)
0 . Letting M be the adjacency matrix of H0, the adjacency
matrix of H
(2)
0 is simply M
2, which implies that the eigenvalues of H
(2)
0 are the squares of the eigen-
values of H0. Therefore, H
(2)
0 is an (n, d, λ)-graph with d = (q + 1)
2 and λ = 2q. Since each vertex of
H
(2)
0 is contained in exactly q + 1 loops, the graph G0 formed by removing these loops has adjacency
matrix M2−(q+1)I, implying that G0 is an (n, d, λ)-graph with d = q(q+1) and λ = q+1. Therefore,
2
by the expander mixing lemma, for all X ⊆ V (G0),
∣
∣
∣
∣
eG0(X)−
q
q2 + 1
(|X|
2
)∣
∣
∣
∣
≤ (q + 1)|X|,
where we used the fact that n = q3 + q2 + q + 1 = (q2 + 1)(q + 1).
Note now that every loop inH0 has exactly q other neighbours with which it can form a non-degenerate
walk of length two. Therefore, since any X ⊆ V (H0) contains at most |X| loops, we remove at most
q|X| edges from X when forming G1 from G0. By the estimate above, this implies that, for all
X ⊆ V (G1),
∣
∣
∣
∣
eG1(X)−
q
q2 + 1
(|X|
2
)∣
∣
∣
∣
≤ (2q + 1)|X|.
Therefore, G1 is (p, β)-jumbled with p = q/(q
2 + 1) and β = 2q + 1.
Recall that G1 is the union of cliques, while G is the union of random bipartite graphs, one for each
clique in G1. Our aim now is to show that asymptotically almost surely G is (p, β)-jumbled with
p = q/2(q2 + 1) = Ω(n−1/3) and β = O(q log n) = O(
√
pn log n). We will do this in a rather naive
fashion, estimating the probability that
∣
∣
∣
∣
eG(X) −
q
2(q2 + 1)
(|X|
2
)∣
∣
∣
∣
≤ Cq|X| log n (1)
for any given X and taking a union bound. To do this, we will need the following concentration
inequality for quadratic forms in independent random variables due to Hanson and Wright [7] (the
exact version we state follows from Theorem 1.1 in [10]).
Lemma 1 Let Z = (Z1, . . . , Zt) ∈ {−1,+1}t be a random vector with independent components each
of which is equal to 1 or −1 with probability 1/2. Let M be a t× t real matrix. Then
P[|ZTMZ − E(ZTMZ)| > ǫ] ≤ 2 exp
{
−cmin
(
ǫ2
‖M‖2F
,
ǫ
‖M‖
)}
,
where ‖M‖F = (
∑
i,j m
2
ij)
1/2 is the Frobenius norm and ‖M‖ = supx 6=0 ‖Mx‖2/‖x‖2 is the spectral
norm.
Suppose now that G[X] is the union of s cliques T1, . . . , Ts, of orders t1, . . . , ts, and let t = t1+ · · ·+ ts.
We define t random variables X1, . . . ,Xt, each equal to 1 or −1 with probability 1/2, and assign one
of these random variables to every vertex of every clique, noting that any given vertex may receive
multiple random variables, but only one relative to any given clique.
Suppose that the random variables assigned to the clique Ti are Zi1, . . . , Ziti . If vi is the vertex whose
neighbourhood in H is Ti, the value of Zij determines whether its corresponding vertex Ti(j) is placed
in Avi or Bvi , with Ti(j) placed in Avi if Zij = 1 and Bvi if Zij = −1. The number of edges in G[Ti]
is then
eG(Ti) = |Avi ||Bvi | =


ti
2
+
1
2
ti
∑
j=1
Zij




ti
2
− 1
2
ti
∑
j=1
Zij

 =
t2i
4
− 1
4
ti
∑
j=1
ti
∑
k=1
ZijZik.
3
Summing over i, we have
eG(X) =
s
∑
i=1
eG(Ti) =
s
∑
i=1
t2i
4
− 1
4
s
∑
i=1
ti
∑
j=1
ti
∑
k=1
ZijZik
=
1
2
s
∑
i=1
(
ti
2
)
+
1
4
s
∑
i=1
ti −
1
4
s
∑
i=1
ti
∑
j=1
ti
∑
k=1
ZijZik
=
eG1(X)
2
− 1
4
s
∑
i=1
∑
1≤j 6=k≤ti
ZijZik.
Therefore, by our estimate on eG1(X), it only remains to show that
Q =
s
∑
i=1
∑
1≤j 6=k≤ti
ZijZik
is smaller than Cq|X| log n with sufficiently high probability.
Let M be the t× t matrix whose entry mjk is equal to 1 if j 6= k and j and k (each of which represents
a vertex associated to a particular clique) are from the same clique Ti. Otherwise, we take mjk to be
0. Then Q = ZTMZ and it follows from the Hanson–Wright bound that
P[|Q| > Cq|X| log n] ≤ 2 exp
{
−cmin
(
C2q2|X|2 log2 n
‖M‖2F
,
Cq|X| log n
‖M‖
)}
.
But it is straightforward to verify that
‖M‖2F =
s
∑
i=1
ti(ti − 1) = 2eG1(X) ≤
|X|2
q
+ 6q|X| ≤ 2max
{ |X|2
q
, 6q|X|
}
and, writing ρ(A) for the spectral radius of a matrix A,
‖M‖ =
√
ρ(M∗M) = ρ(M) ≤ sup
x 6=0
‖Mx‖∞
‖x‖∞
= max
1≤j≤t
t
∑
k=1
|mjk| ≤ q.
Therefore,
P[|Q| > Cq|X| log n] ≤ 2 exp
{
−cmin
(
1
2
C2q3 log2 n,
1
12
C2q|X| log2 n,C|X| log n
)}
≤ 2 exp{−2|X| log n}
for C sufficiently large in terms of c. Applying the union bound, we see that the probability there
exists a set X such that (1) fails is at most
n
∑
|X|=1
(
n
|X|
)
2e−2|X| logn ≤ 2
n
∑
|X|=1
n|X|e−2|X| logn ≤ 2
n
∑
|X|=1
e−|X| logn ≤ 2
n− 1 .
The result follows.
4
3 Concluding remarks
Our construction also extends to give a sequence (ni)
∞
i=1 of positive integers such that, for each
i ≥ 1, there is a C5-free graph Gi on ni vertices which is (p, β)-jumbled with p = Ω(n−3/5i ) and
β = O(
√
pni log ni). The construction starts with the C10-free polarity graph of Lazebnik, Ustimenko
and Woldar [9], but follows the proof of Theorem 1 in all other respects. Because we lack optimal
constructions of C2ℓ-free graphs for ℓ ≥ 6, our method does not extend further to give constructions
of pseudorandom C2k+1-free graphs for k ≥ 3. As mentioned in the introduction, this is a distinct
weakness of our method when compared to Alon’s, which does extend to longer odd cycles [2, 8].
An alternative method for proving the jumbledness of our construction G might be to estimate the
eigenvalues of its adjacency matrix M by using the fact that Tr(M2k) is both the sum of the 2kth
powers of its eigenvalues and the number of walks of length 2k in G. However, G is constructed
by starting from a graph G1 which is a union of cliques and then taking a random bipartite graph
within each clique. This process causes an imbalance between odd and even cycles within each clique,
deleting all odd cycles, while doubling the proportion of even cycles relative to the density. This makes
it difficult to count the number of degenerate walks of length 2k without having close control over the
counts of degenerate walks of different types in the base graph G1. Nevertheless, it is plausible that
this could be done, and may even allow one to save the lost logarithmic factor in β.
Another definite weakness of our method is that it is not explicit and so, unlike Alon’s example, cannot
be used to give a constructive lower bound for the off-diagonal Ramsey number r(3, t). It remains
to decide whether there is some more explicit method for choosing large bipartite subgraphs of the
cliques in G1 which also produces a highly pseudorandom subgraph.
Acknowledgements. I would like to thank the anonymous referee for a number of helpful remarks.
References
[1] N. Alon, Explicit Ramsey graphs and orthonormal labelings, Electron. J. Combin. 1 (1994), R12,
8pp.
[2] N. Alon and N. Kahale, Approximating the independence number via the θ-function, Math. Pro-
gram. 80 (1998), 253–264.
[3] T. Bohman, The triangle-free process, Adv. Math. 221 (2009), 1653–1677.
[4] D. Conlon, J. Fox and Y. Zhao, Extremal results in sparse pseudorandom graphs, Adv. Math. 256
(2014), 206–290.
[5] P. Erdős, M. Goldberg, J. Pach and J. Spencer, Cutting a graph into two dissimilar halves, J.
Graph Theory 12 (1988), 121–131.
[6] P. Erdős and J. Spencer, Imbalances in k-colorations, Networks 1 (1971/72), 379–385.
[7] D. L. Hanson and F. T. Wright, A bound on tail probabilities for quadratic forms in independent
random variables, Ann. Math. Stat. 42 (1971), 1079–1083.
[8] M. Krivelevich and B. Sudakov, Pseudo-random graphs, inMore Sets, Graphs and Numbers, Bolyai
Soc. Math. Stud., Vol. 15, 199–262, Springer, Berlin, 2006.
5
[9] F. Lazebnik, V. A. Ustimenko and A. J. Woldar, Polarities and 2k-cycle-free graphs, Discrete
Math. 197/198 (1999), 503–513.
[10] M. Rudelson and R. Vershynin, Hanson–Wright inequality and sub-Gaussian concentration, Elec-
tron. Commun. Probab. 18 (2013), 1–9.
6


A sequence of triangle-free pseudorandom graphs

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