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2325b8ed |
Erdős's Alice–Bob clique game on $K_n$: does Bob have a winning strategy for all $n\geq 3$? (Erdős #778) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
0e1e781a |
Erdős–Rogers problem: largest triangle-free induced subgraph forced in a $K_4$-free graph (Erdős #620) |
OPEN |
0 inv |
4.0 |
1.5 |
36d ago |
|
d6b3ed12 |
Pin down $t(r)$: transversal number forced by a local $\tau\leq 1$ condition on $r$-uniform hypergraphs (Erdős #616) |
ACTIVE |
3 inv |
3.0 |
2.0 |
15d ago |
|
40f3f739 |
Determine $f(n,k)$: fewest edges forcing degree $\geq k$ in every $(k+2)$-vertex induced subgraph (Erdős #614) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |
|
c6a367a7 |
Diameter of $K_{k+1}$-free graphs with minimum degree $d$: is it at most $(3-2/k)n/d$? (Erdős #612) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
eb2b00ba |
Sublinear clique transversals under a large-clique hypothesis: is $\tau(G)=o_c(n)$? (Erdős #611) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
54592ee7 |
Edges forcing an $r$-triangle edge: are the thresholds $e(n,r)$ asymptotically flat in $r$? (Erdős #600) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
c6a62326 |
Clique transversal vs. independence: is $\tau(G)\le n-H(n)$ for all graphs? (Erdős #151) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
9d8169b1 |
Strong chromatic index conjecture: is $\mathrm{sq}(G)\le\tfrac54\Delta^2$ for every graph? (Erdős #149) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
6907909b |
Turán density of $C_4$ in the hypercube: does $(1/2+o(1))n2^{n-1}$ edges force a $C_4$? (Erdős #86) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
c7e81a65 |
Is $f(n)$ — the min-degree threshold forcing a $C_4$ — eventually monotonic? (Erdős #85) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
781d464a |
Force a large regular induced subgraph: does $F(n)/\log n\to\infty$? (Erdős #82) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
ac3b55c4 |
Partition the edges of a chordal graph into cliques: is $n^2/6+O(n)$ always enough? (Erdős #81) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
63e94a95 |
Bound $c(n)$, the least $k$ past which an $n$-cube splits into $k$ homothetic subcubes (Erdős #769) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |
|
846c7138 |
Self-avoiding walk displacement: does $d_2(n)/\sqrt{n}\to\infty$ and $d_k(n)\ll\sqrt{n}$ for $k\geq3$? (Erdős #529) |
OPEN |
0 inv |
3.5 |
2.0 |
36d ago |
|
745418e0 |
Chromatic number of the plane (Hadwiger–Nelson): pin $\chi(\mathbb{R}^2)$ between 5 and 7 (Erdős #508) |
OPEN |
0 inv |
4.5 |
2.5 |
36d ago |
|
c43c5eec |
Smallest $k$: 2-colour the plane with no red unit pair and no blue unit-spaced $k$-AP (Erdős #188) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
a41287a4 |
Characterise the Ramsey finite point sets in Euclidean space (Erdős #174) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
38f9bab2 |
Monochromatic triangles under any 2-colouring of the plane: at most one exceptional shape? (Erdős #173) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
25026bec |
Common finite-chromatic subgraph of two graphs of chromatic number $\aleph_1$ (Erdős #62) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
b83472a9 |
Erdős–Hajnal conjecture: does an excluded induced $H$ force a polynomial clique or independent set? (Erdős #61) |
OPEN |
0 inv |
4.5 |
1.0 |
36d ago |
|
997fb055 |
Points in $\mathbb{R}^d$ forcing $n$ with all pairwise distances distinct: is $f_d(n)=2^{o(d)}$? (Erdős #1088) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
3d9e309e |
Estimate $h(n)$: distinct-radius circles forced through triples of $n$ planar points (Erdős #831) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
d8aac4b1 |
Determine $n_k$: fewest general-position points forcing $k$ whose triples give all-distinct circle radii (Erdős #827) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
2bffc76c |
Generalized orchard problem: determine $\lim F_k(n)/n^2$ and $\lim f_k(n)/n^2$ for $k$-rich lines (Erdős #669) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
9b81043f |
For which $n$ can some triangle be cut into $n$ mutually congruent triangles? (Erdős #634) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
f152506a |
Max number of $k$-rich lines when no $k+1$ points are collinear: is $f_k(n)=o(n^2)$ for $k\ge4$? (Erdős #588) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
4acb7a22 |
Determine the self-avoiding-walk connective constant $C_k$ in $\mathbb{Z}^k$ (Erdős #528) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
0461cec7 |
Finitely many perfect powers (and powerful numbers) among sums of distinct factorials? (Erdős #1108) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
fcbfbcfd |
$p$-adic valuation of sums of distinct factorials: bound $f(a,p)$ or force it to infinity (Erdős #404) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |
|
98ac231c |
Determine the average order of $g_k(n)$, the factorial-excess with $a_1!\cdots a_k!\mid n!$ (Erdős #400) |
OPEN |
0 inv |
2.5 |
3.0 |
36d ago |
|
5e58fb07 |
Is there a threshold $c$ so every planar set of measure $\ge c$ contains a triangle of area 1? (Erdős #352) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
19cf0236 |
Must every infinite bounded-step walk in $\mathbb{Z}^3$ contain three collinear points? (Erdős #193) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
6a5dfe5c |
How many unit circles can $n$ points determine through $\ge 3$ points? Prove $o(n^2)$ (Erdős #104) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
225b1e1b |
If $cn^2$ lines each hold $>3$ of $n$ points, must some line hold $h_c(n)\to\infty$? (Erdős #102) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
b312bc8a |
How many 4-point lines can $n$ points with no 5 collinear span? Prove the count is $o(n^2)$ (Erdős #101) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
db8fe33c |
Growth of $\tau_\perp(n)$, the count of coprime consecutive divisors of $n$ (Erdős #1100) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
e788985a |
Divisor sums of irreducible polynomial values: is $\sum_{n\le X}\tau(f(n))\sim cX\log X$? (Erdős #975) |
OPEN |
0 inv |
3.5 |
2.0 |
36d ago |
|
65c0dcd3 |
Does the ratio $f(2n)/f(n)$ tend to a limit, where $f(n)=\sum_{k\le n}\tau(2^k-1)$? (Erdős #893) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
f8372cc1 |
Is the number of divisors of $n$ in $(\sqrt n,\sqrt n+C n^{1/4})$ bounded by an absolute constant? (Erdős #887) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
17c8d8c1 |
Bound the number of divisors of $n$ in $(\sqrt n,\sqrt n+n^{1/2-\epsilon})$: is it $O_\epsilon(1)$? (Erdős #886) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
4bbd96c3 |
Least spread $f(n)$ of a factorization of $n!$ into distinct integers (Erdős #393) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
5c91f14c |
Factor $n!$ into distinct parts $>n$: does $f(n)-2n\sim c\,n/\log n$? (Erdős #390) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
62039dc0 |
Degenerate 4-point subsets (a repeated distance among the six): is the count $n^{3+o(1)}$? (Erdős #1087) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
96a01429 |
Unit-area triangles: how many triangles of the same area can $n$ planar points span? (Erdős #1086) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
808f7b54 |
Unit distances in $\mathbb{R}^d$: estimate $f_d(n)$, the maximum number of unit-distance pairs (Erdős #1085) |
OPEN |
0 inv |
4.0 |
2.5 |
36d ago |
|
b04adb41 |
Contact number problem: max unit-distance pairs among $n$ points pairwise $\geq 1$ apart (Erdős #1084) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
b1ac53e8 |
Distinct distances in $\mathbb{R}^d$: is the minimum $n^{2/d-o(1)}$ for every fixed $d\geq 3$? (Erdős #1083) |
OPEN |
0 inv |
4.0 |
1.5 |
36d ago |
|
289a846a |
Largest gap between the top two distance multiplicities of an $n$-point planar set (Erdős #959) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
17d49f34 |
Do $n$ points whose pairwise distances differ by at least 1 force diameter $(1+o(1))n^2$? (Erdős #670) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |